Sunday, January 30, 2011

Topology and the Moore Method--i.e. what I'll blog about for the next few weeks

  So I'd like to start blogging about this class I'm teaching.  I decided to use the Moore Method to teach my Topology course.  I'll start by giving a little background.  My first and favorite math courses were taught by Dr. Phil Tonne and Dr. Bill Mahavier at Emory University.  Dr. Tonne taught me Matrix Algebra and Calculus, and Dr. Mahavier taught my Real Analysis class.  I was a teenager when I took these classes and the format of the classes spurred me to work for hours at a time on math at the kitchen table with my mother occasionally putting food in front of me or telling me it was time for ballet class.  What I wouldn't give to get back to the level of focus and dogged determination I had at that time!
   Anyway, I got up to the board in those classes, sharing solutions to problems in Calculus class, and proofs in Real Analysis, and I still remember some of the proofs and ideas I learned in those classes after fifteen years.  Most of all, I remember the feeling of accomplishment and excitement that I got from doing some really hard problems myself...And I didn't really take very many notes either...I went on to take some time off from math in my Junior and Senior years of high school when I turned my attention to French, Latin, and Linguistics, then back to math in college, and again in graduate school.  In college, I had more amazing Texas-style classes with Dr. John Neuberger, who made me think that I might study Differential Equations simply because his class made it seem so great.  But ultimately I was drawn to Topology, the field of the illustrious Dr. R.L. Moore, after whom the method of teaching I am inspired by is named.
   So that's why I'm inspired to teach by the Moore Method.  It was so influential in my own pursuit of mathematics that I feel it might stoke the fire of others who are already somewhat interested in mathematics.  And since teaching this way means that I don't get to talk much in class since the students are doing most of the talking, it means that I should talk here instead :).
   Being already about two weeks into the course, I'll tell you where we are right now.  We have made it almost all the way through about nine pages of notes, which briefly cover the Topology of the real line, the general definition of a Topology,  exercises on what makes two topologies the same, the separation axioms and their relationships to one another, many theorems and false statements concerning boundary, limit points, and interiors of sets in topologies satisfying various separation axioms.
    The students present around two or three exercises, counter examples, or proofs per class meeting, but only about half the students (out of 20) have made it to the board so far.  There has been quite a bit of juggling of the roster as students drop and add.  I started with 28 students signed up.  After listening to my opening spiel the first day, about five of them dropped immediately.  So I suppose that this scenario could have been a lot worse if I'd started with only those five students.
   The logistics of the class are a little hard to stick to as I sometimes get excited and forget to give those who have not yet presented a chance to put something up.  Instead, I've forgotten and asked for volunteers, which is honestly not seeming like such a good idea since I'll get the same five or ten kids the whole semester I suspect.  I have been good about going to the back of the room and being part of the audience, but there is some reticence to ask questions amongst the students.  So I've been having to sort of drag it out of them by calling on people at random to summarize the work of their peers.
    I'm really wondering how this would work in my introductory Calculus course.  I feel like it would be worthwhile since I'm writing their first exam and have only a vague idea of what anyone REALLY knows.  But they might hate me... Too late to change the format of the class now.  One thing I've learned in teaching:  stick to one thing and make it work.

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