Friday, April 04, 2014

How I learn a result

There is a theorem I want to learn. How do I go about it? (How do YOU go about it?) I give an example here which also leads to  pointers to some mathematics of interest.

A division of a cake between n people is PROPORTIONAL (henceforth prop.)
if everyone thinks they got (IN THEIR OWN MEASURE- AND PEOPLE CAN DIFFER WILDLY) at least 1/n. A division of a cake is ENVY FREE if everyone thinks they got the biggest (or tied) piece.

There are two kinds of protocols- Discrete and Moving Knife. I discuss Discrete here.

I had read several books on fair division (see here for a review) and had some interest in the topic. In particular (1) I knew the (fairly easy) discrete 3-person Envy Free protocol, but did not know (2) the rather complicated discrete 4-person Envy Free Protocol. And I wanted to learn it! How to do I do that (this isn't quite advice on how YOU should do it, but I am curious what you think of what I do AND what you do.)

  1. Find a good source for it. I used the original article from the American Math Monthly. It is NOT always the case that the original source is the best (I discussed this in another blog post.)
  2. Create a REASON to learn it with a deadline. I both agreed to teach it in seminar AND I volunteered to do an HONORS COURSE (interdisciplinary) on Fair Division (nickname: Cake Cutting). The first time I taught the course I didn't get to the theorem (getting my feet wet in both HONORS courses and the material was a limiting factor) but I have in subsequent years.
  3. While reading make HANDWRITTEN notes for the lecture I am going to give. I try  to understand everything that I  write down before  going on, but sometimes, I have to go forward and then backward. I find myself redoing parts of the notes many times. The notes  have many examples, pictures (which is why I don't type them initially), and counterexamples to check that the premises are needed. I check EVERY detail. What I present will  be less detailed.
  4. Type in the notes. (mine for 4-envyfree are here ) For me, somehow, being forced to type it  in I find corrections or better ways of saying or doing things. Also this produces notes for students or others. This MIGHT over time lead to survey articles or even textbooks but I DO NOT think in those terms. The notes that are produced are VERY GOOD FOR THE STUDENTS IN THE CLASS, or THE PEOPLE IN SEMINAR but NOT all that good for anyone else. (This may be why some textbooks are crap--- it is hard to turn class-specific notes into something anyone can use.)
  5. TOSS OUT the handwritten notes. Knowing you will do this makes the typewritten version better. Also its hard to keep track of where one puts handwritten notes. I used to keep a math notebook but its hard to find anything in it.
  6. In subsequent years when I teach the material I take my typewritten notes and make handwritten notes out of them. This force me to refresh on the material and I teach better if I can glance at notes that use large letters and symbols. If I do the course often enough I no longer need to. When I taught the Fair Division course in Fall 2013 I could do this proof off the top of my head.
There are some variants, Pros, and Cons of the above system:

  1.  If the material is easy I may go straight to type and skip handwritten. This was the case for protocols for proportional  cake cutting.
  2. I sometimes translate my notes, NOT to typed notes but to slides. There are some theorems whose ONLY writeup are my slides. I may or may not ever get them into notes  (that's a tautology).  I'll mention two but note that, even more than notes, slides are not really a good read unless you are taking the class. They are (1)  An adaption of Mileti's proof of the infinite Canonical Ramsey Theory to the finite case- it gives pretty good bounds, though not the best known. Very good for teaching, so I may submit to Journal of  Pedagogical Ramsey Theory.  I am sure these slides contain a correct proof since I ran them by Mileti himself., who was surprised and delighted  that someone got around to finitizing his proofs. The slides are here and here.(2) Using linear programming on some cake cutting problems. I doubt this is new, though I could not find it in the literature. I just did it for my class. Slides are here.
  3. If my GOAL is stuff for my class I may end up giving up. For example, I sort-of want to read the LOWER BOUND of Omega(nlogn) on number-of-cuts for n people, on prop. cake cutting due to Jeff Edmonds and Kirk Pruhs (this is Jeff Edmonds paper page). But the proof looks just a bit too hard for my students (recall- they are NOT math people, they are BRIGHT students from across majors). Knowing that I won't get to teach it to them I didn't quite have the motivation to look at it. I probably will someday--- for seminar.
  4. If its not just one paper but an entire area of knowledge I try to find all the papers in the area and make a website out of them. This is easier in a limited area. My Erdos Distance Problem Website has only 10 papers on it (its prob missing a few), where as my Applications of Ramsey Theory to TCS websiteApps  has 63 papers on it (I am sure its missing many). 
  5. One can get carried away with GATHERING papers as opposed to READING them.
What if you want to read a result, you begin, and its rather hard and/or you need a lot more background? There are several competing schools of thought:

  1. If at first you don't succeed, Try Try Again (credited to  William  Edward   Hickson).
  2. If at first you don't succeed, give up, Then Try Try again. Then quit. There's no point in being a damn fool about it  (credited to W.C. Fields)
  3. If at first you don't succeed then destroy all evidence that you tried.
  4. If at first you don't succeed, skydiving is not for you.
The question of when to hold 'em, when to fold 'em, when to walk away, and when to run permeates life and mathematics as well as gambling.

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