Here is a short report on the big experiment for this term, and a related note on a realization from today with wider applicability. I expect that this will start well, and then ramble on as I fiddle with some ideas.
I am teaching iteration number ten of my Modified Moore Method Euclidean Geometry course. This semester I am making an effort to refocus on the basics: managing and mentoring the students as much as I can.
At this point, my theorem sequence is very stable. (I am no longer surprised much by what happens in this course.) This allows me to work on the other aspects of the course. I feel like I have started to let some important things go in the last year or so, and now I want to sharpen up. What has been lacking? I don’t think I have kept on top of the students to keep them engaged as well as I might. And I don’t think I have done a good job selling the method of instruction, either.
So, I was much more deliberate about introducing myself to each of my sixteen students on the first day. I have been very explicit about my expectations and my willingness to help them meet those expectations (which are rather high). And I will be making a conscious effort to check in with as many students as possible each day.
The first week was a rousing success, I think. Each day we got at least one theorem. We have already set the expectation for what counts as an argument. (Well, surely, there is still some work to be done.) The class has made two conjectures. We took some time to discuss some basic points of what acceptable writing will look like. I even successfully navigated our first potential difficult situation and found something positive in it. All in all, I am feeling pretty good about this.
I think our next test comes when we have to finish conjecture 1.1. They haven’t addressed the second statement in that, yet.
And sometime next week I will have to steal ten minutes to talk with them about my Standards Based Assessment experiment for the term.
…continuing from the previous post. Here are some other things I am thinking about.
- It is good to have a crowd. That is, it is good to have a group of like-minded people to lean upon. I missed that the last year or so.
- I need to refocus on intentionality in my teaching. I have made many sloppy, quick decisions lately. I want to be sure that the choices I make are being made for reasons.
- The trouble of finding a decent assessment method is killing me. Until I find some sort of partial resolution to this, I will not make any other progress as an educator.
- I want to refocus on mentoring students personally. I have gotten away from this. It has always been the case that students express fear about talking with me. I don’t quite know where this comes from, but it has always been so. In the last few semesters I haven’t done enough to counteract this. Students aren’t coming to visit me and discuss things one-on-one. This hampers my ability to help them.
- I need to sell what I am doing. I am not doing that enough.
With these in mind, I wrote some things I might consider saying to a hypothetical student.
Embrace Challenge. I am here to make sure you get stuck and help you get past it. This is the only way I know to foster true intellectual growth.
Math is hard. I am just laying that bare instead of hiding that behind slick lectures.
My intention is to give you many opportunities for growth and learning. I will try to put you in a situation where success is possible. But I will not put you in situations where success is guaranteed. There is no growth, no learning, no empowerment, in finishing a task you are sure of completing before you begin. So you will often feel unsure of how to succeed, at least at first, and maybe for a lot longer than you are accustomed. I am not going to guarantee your success, not on the small scale of a single task, nor on the larger scale of this whole course. But I am confident in your success. I believe deep in my bones that each student can eventually prosper here. My job is to provide you with guidance so you can grow and succeed in ways you were not capable of before.
Originally Posted 01-16-2012
This semester I am supervising an MA student. This is not a full-blown MA thesis; instead, the student has chosen the “non-thesis option” for his work, which requires him to do 2 cerdit hours worth of research work and write a paper. I talked to a few of my colleagues, and it seems that the requirements are a bit nebulous, but certainly don’t include new work.
I talked with the student and he is interested in geometry, has a background in art (some computer animation), and is curious about fractals. Also, he did a summer project with me a while back, and learned a lot about the geometry of the hyperbolic plane. So, I pulled my copy of Indra’s Pearls down off the shelf…
Now, I haven’t read the book before. (One of my vices is buying math books that I want to read “someday.”) Easy selling points here were that the book is full of pretty pictures, and one of the co-authors is my academic grandmother.
I figure that if a student is going to do some research, they have to make something. In this case it is not a new theorem, or a new argument for an old theorem, but instead a picture. Rather, I am going to ask him to make some pictures of limit sets of Kleinian groups.
If all goes well, he will actually write a bunch of routines in Python/Sage/Cython for making such pictures that can be used for a variety of groups. I don’t know if this has been done before (I suspect it has), but I don’t really care. The student will be doing something of academic value, and it will help me learn some stuff that I really should have finished absorbing by now.
So today I read several chapters of the book to get rolling. It starts off very gently. The book is written for a mathematical layman, really, so they start with chapters introducing complex numbers, symmetry, groups, and Moebius transformations of the Riemann sphere. There is pseudo-code weaved in, too, so that a reader can get going with some basic drawings.
It looks like the rest of tonight’s reading will be about Schottky groups. I haven’t seen anything new (to me), yet, but several of these basic concepts are things that I find myself explaning to students often, and I might adopt one or two turns of phrase.
The goal I set for the first week was to try to understand how the group is essentially the set of isometries of hyperbolic 3-space. I am sure that will keep him occupied for this week. Next we can try to draw pictures of the action of individual Moebius mappings
Oh, if you want, here is a link to the book.
Originally Posted 01-09-2012
Well, that went smoothly enough. I managed to make it through the day without any major troubles, despite my incomplete preparation for linear algebra.
My new toys were waiting in my office on Saturday night when I arrived back in Cedar Falls. I got to use the iPad in all three classes to take notes and pictures of my students, and I used the MacBookPro to show off the Sage notebook server for matrix computations in linear algebra. I was very pleased. I am still working on set-up, but I have new browsers and LaTeX is up and running.
So how did classes go?
We had good success getting through several of the modeling problems, and we had a nice discussion about the difference between a “time-parametrized model” where exponential growth is determined by an exponential population function, and the dynamical system approach where the same model is determined by a linear function as dynamics. They have new problems for Wednesday, which emphasize the idea of a fixed point.
We spent our time talking about linear combinations of matrices. I asked the students to get a copy of the first few chapters of the Woodruff-Grout text. I showed them how to do some very simple stuff in Sage. We will talk a bit about vectors and geometry some on Wednesday.
I still need to set up my WeBWork course, get my Sage intro up and running, and update my personal web page’s linear algebra section. I think this will be my day tomorrow.
Mathematics in Decision Making
This went really well. We had time for the student to work on all five problems from today’s activity and we discussed the first four. I think the concept of checking that two sets are the same size comes from a “matching” between the two sets. Kyle was a good help, too. I am very pleased to have a TA to deal with a slightly larger class.
Well, it seems like I have a lot to do on this front this semester. I spoke with all four potential students today about scheduling.
One graduate student with a non-thesis option will be studying making fractal limit sets on the sphere at infinity for Fuchsian groups and their friends. He is inspired by the book Indra’s Pearls by Mumford, Series and Wright.
One graduate student starting his thesis work by studying how to think of closed orbits of the geodesic flow of the modular surface as knots in the 3-sphere minus a trefoil knot. (Basically following Ghys’ ICM address from 2006.)
One undergraduate continuing a project on the divisibility of m-ary parition numbers. Really, this is about some strange structural polynomials that appear when studying this problem. We have experimental results, and we need to find a proof.
Possibly a new student to study some discretized version of curve shortening flow. We could start with the Monthly Paper by Chow and Glickenstein (2007).
I didn’t take any time to work on committee work. I have to schedule some meetings to get rolling on our search for funding to support undergraduate research efforts.