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.

EQUATIONS are now a monthly staple in Wired

 A great teaching tool is now appearing every month in the media!  Starting back in May, mathematics writer Julie Rehmeyer started writing a new column in Wired Magazine.  An entire page features one equation along with a fabulous graphic (much better than the silly one I made to the right) to help bring it to life.  There are even sliders on the variables to help readers understand the range of possible scenarios modeled by the equation.

May: Carbon Emissions
June: Phantom Traffic Jams
July: 3-D Rendering
August: Power of Waves
September: Roller Coasters

It hasn't been that long since my last post

    It's been about 7 months, maybe 200 days or so, since my last post.  And while you may not have been counting the 4800 hours, you probably aware of the 5 million barrels or 210 million gallons of oil that were being dumped into the gulf.  This recent SEED slide show highlights the our deficiency as humans when it comes to comprehending these sorts of massive numbers.  Perhaps this is part of the reason we cannot stop buying drinks in disposable containers.  The images in the slide show come from Chris Jordan's 2009 book entitled Running The Numbers .  Of course, the idea of using technology and imagery to help us wrap our minds around the gargantuan nature of our world is nothing new.  The short video Powers of Ten from the 1960's highlights the wonders of the universe by expanding our field of view by one power of ten every 10 seconds.  In the video, we see a square that is 10^8 meters on a side framing the earth and a square that is 10^-8 meters per side framing a coil of DNA.

   Recently, while teaching my Math 124 Calculus Course, I came up with a little related rates program to help the students wrap their minds around the spreading of the oil in the gulf coast.  According to a NY times article from this summer, anywhere from 12 to 25 thousand barrels of oil per day were being dumped.  Wikipedia's site on oil slicks asserts that an oil slick is no thicker than about .002 millimeters.
There are 42 gallons of oil in a barrel.  So we will go with the round and reasonable number of about 1 million gallons per day.  A quick conversion gives us

(1 millions gallons/day) (3.79 liters/gallon) (10^6 mm^3/liter) = 3.79 (10^12) mm^3/day

Now we make a few fairly broad assumptions:

1. The oil spreads in a circular manner and always has uniform thickness. 
2. The oil is being spilled at a constant rate.

Most of us have experienced that as liquid spills out onto a surface, it spreads quickly at first and then slows down over time even if we do pour at a constant rate.  But how exactly is the rate of change of the radius of the spill related to the current radius of the spill?

We now answer that question with a little calculus.
Suppose V(t) denotes the volume of oil spilled at a time t where t is measured in days.  According to the assumptions we made,
V(t)=pi* r(t)^2 h where r(t) is the radius of the slick in millimeters at time t (in days) and h is the thickness of the slick.  So, using the chain rule, we have V'(t)= 2pi*h*r(t)*r'(t).
Because we know that the rate of change of volume of the oil spilled is constant at 3.79 (10^12) mm^3/day, we can determine the rate of change of the radius with respect to time in terms of the radius of the slick at a particular time.  So we see that r'(t)*r(t) is approximately 302(10^12)mm^2/day.  If we choose to measure r(t) in kilometers, then r'(t)*r(t) is approximately 302 km^2/day.
In other words, the rate of change of the radius of the slick is inversely proportional to the current radius.
So, when the slick is 1 km in radius, it is spreading at a rate of 302 kilometers per day.
But when the radius is 302 km, the radius is changing at a rate of only 1 kilometer per day.
By the way, 302 km is about 188 miles.  But the rate of spread is slowing, so how long would it take for the oil to reach shore if the spill occurred 100 miles off shore?  If the rate of the spill really does stay constant (i.e. no successful clean up) then we see can approximate r(t) as 25 t^(1/2).   When is 100=25t^(1/2)?  After about 16 days. So a major oil spill, even if it occurred 100 miles out into the ocean could reach the shore in a few weeks!
   I can only hope that the spread of information is as successful and uniform.  And I also make the observation that I waited about the same number of days to post as the number of kilometers that the oil slick would have grown in even one day.

P.S. How accurate is this little estimate?  Certainly we made many simplifications and overlooked all clean up efforts.  Looking at an interesting app from the NY times, I see that it did indeed take about three weeks for the spill to be noticed on the shore of Mississippi about 100 miles away from the source.

The Last Year of Grad School

  Nobody tells you about the emotional part of leaving grad school.  The time I've spent here has been longer than the time I've spent anywhere outside of the town I grew up in.  I've developed relationships here that will (I hope) last me a lifetime.  I have become a wife here, an adopted "auntie", and a teacher to many.  As I bike into my apartment complex, I wave left and right at familiar faces.  I see children who were babies when I started grad school here.  I stop to talk with former students who are themselves moving on to grad school, new jobs, teach for america, or travel abroad.  Getting ready to go isn't just about writing my thesis -- it's about saying goodbye to the people and places I've bonded with.
   Granted, I'm happy right now to be finishing the umpteenth grade and to be starting a new chapter in my life.  But I'm just warning you -- steel yourself against the fact that you will be processing a lot more than just your mathematics as you write up that thesis.   You'll be thinking about what  to keep and what to toss, you'll be spending time with friends, reconnecting with those you've lost touch with, visiting with those who are just near enough to drive to see and just far enough to not bump into.  You'll be wondering if anyone else will carry on the traditions you've started.  You'll see your surroundings with different eyes.  Try it now...It might be an interesting exercise.

Manifold Fashion

   This week Dai Fujiwara, a fashion designer, presented a collection based on Topology, specifically Fujiwara discussed his collection with Professor Bill Thurston, who is famous for his Geometrization Conjecture .  Here is a blog entry with pictures that discusses the role of geometry in this years fashion.  Another designer is also mentioned as incorporating geometric ideas in the NY Times.  Also see the YouTube interview of Thurston and Fujiwara .  

Census and Sensibility

    With the 2010 Census about to take place in the middle of this month during what is Spring Break for many college students, we might start to think about fairness.  As people receive so much junk mail, it's important to convey the importance of filling out the ten questions (which is the fewest number to ever appear on a census).  According to the Census2010 website $400 BILLION is distributed according to the numbers recorded in the census.  Just recently, a house committee decided to table a proposal for a bipartisan commission to help decide the redistricting that will occur when the census is over.  So how will redistricting be decided fairly by those people who have themselves benefitted from possibly unfairly drawn districts?

    Here we see the 12th congressional district that was approved in 1994, and which was controversial for being racially "gerry-mandered" .  Now, we have to ask ourselves if there is anything wrong with a district having an "bizarre" shape, whether this is somehow giving an unfair advantage.  Perhaps requiring that districts be "sensibly shaped" in some way is a reasonable way to ensure fairness.  Perhaps not.

Let's also take into consideration that "There's nothing Maryland can do about its bizarre shape," as economist Christopher Chambers said at the AAAS symposium on Fairness and Mathematics.  Professor Chambers is one of the authors of an academic article entitled "A Measure of Bizarreness"  that will soon be published in the Quarterly Journal of Political Science.  So however we may choose to answer the question of "How Bizarre is that shape?" it must take into consideration that the shape may already live in a somewhat bizarre larger one.  The way to measure this according the the authors is to use a path-based measure of convexity.  Given a district, its bizarreness is determined by the probability that it contains the shortest path within the state that joins two randomly selected points in the district.  The higher the probability, the lower the bizarreness.

Here's a January article from Slate magazine concerning this matter.

     

What's Massive about Media?

    Media is the plural of medium, and is a substrate through which information flows.  During the AAAS Mass Media Luncheon, speaker Dr. Jeffrey Kirsch , the Executive Director of the Fleet Science Center , asked us for our reactions to the phrase coined by Marshall McLuhan in the 60's: "The medium is the message."  Just the day before in the Counter-terrorism Symposium, Keith Devlin spoke about the difficulty of quantifying "information" and its reliability.  These discussions led me to think about information like light -- a pure and difficult to measure substance whose appearance is determined by the substance through which it passes.  Rather than thinking of information as being stuck in some sort of box to which an elite few hold the key, I recognize that information is difficult to confine.  Just as we created the light bulb, neon signage, fiber optics, and lasers to channel light which is typically free to move about, we design means by which we channel information so that it can be viewed, touched, heard, smelled, and sensed as we see fit.
    We control some of the media through which information passes: our own mental framework, a blog, a podcast, an imax movie, a newspaper article, our social interactions.  But we don't have nearly as much control over the information itself, which is floating everywhere around us in more or less dense and disorganized clouds.  So people who choose to be involved in media are attempting to corral information, this unruly light-like substance, so as to harness some of it's power and help others use it as a tool to brighten their lives.
     With this metaphor in mind, I see that there is power in heterogeneity. With a variety of media, what remains to be seen is which media will excel at fulfilling which roles.  While some media will replace others (like compact fluorescents are increasingly replacing incandescent bulbs), many will coexist or work in concert.
   Still there is the concern that people will be blinded by so much information, as if we are in a world covered in a thick blanket of pure white snow.  Well, I think that's what sunglasses are for!  In other words, people will have to squint for a while before they realize which media they need in order to function, and many members of the public are probably in that squinting phase right now.  So when they realize that they need a way to filter out the intensity in order to focus in on some of the details and survive, they will be scrambling to find good journalists.