Almost everyone in the class is either engaged in thesis research or expects to engage in thesis research very shortly. The way you write up your research for your thesis will depend a bit on the subject area, but it will almost inevitably follow what I call the "P to P+1" Model.
One takes paper P and copies the bibliography except for the addition of a few more recent works. One reprises with modest variations the motivating paragraphs of paper P, and, most important, one explains how P+1 is new in comparison to P. One then does some honest work. Finally, one writes a conclusion that reminds the reader of the larger context for this work and delicately suggest work that may come later.
With few (but regular) exceptions, both good papers and bad papers follow this pattern.
Almost all mathematical progress comes from such evolutionary increments, and one cannot complain. Hadamard's proof of the prime number theorem was "proof mined" by Dirichlet in his study of primes in an arithmetic progression, and the world is better off for Dirichlet's efforts. What we call Central Limit Theory is an amalgam of hundreds of papers that followed the P to P+1 pattern.
Here I encourage you to try something out of the usual mold.
You can forget about the introduction. You can forget about anything complicated --- or general, or new. You don't have to sell anything in the conclusion. Your bibliography is not a genealogy of the subject; you just cite what you use.
You can focus on being interesting and clear.
My bet is that if you do this once early in life, then when you go back to the "P to P+1" Model (as we all must do), then you will go back a stronger and wiser person. You will even have the chance of writing the kind of papers that people read, remember, and thank you for having written.
I am decently confident that the best way for you to write a report that will be genuinely interesting to another reader is for you to begin with a simple problem.
The problem should be posed in the simplest way that you can possibly pose it, yet the problem should be one that you do not expect the reader to be able to solve right away.
Example 1: Show that in any sequence of n^2+1 real numbers, there is either a monotonic increasing subsequence of length n+1 or a monotonic decreasing subsequence of length at length n+1.
Example 2: Show that in any graph with k^2+1 edges, there are either k+1 disjoint edges or there is a "star" with k+1 edges.
You can then give a picture or a numerical example to make sure that your reader understands the problem completely.
If you like you can also give a one or two line pep talk. For the second example, you can honestly say "The solution to this problem leads to generalizations that have had far reaching applications in complexity theory."
Next you coach the reader into seeing how one can solve this problem.
Once the first solution is complete, you propose the "next problem," make sure it is understood, and then you engage its solution. You repeat this same process until you have honestly and completely coached your reader into understanding a sequence of facts and ideas that are genuinely worth knowing.
If you follow this model, you have a much smaller chance of deluding yourself about what is interesting or not. You also can see what insights are valuable versus what might just be technical clutter.
Also, by giving the solutions as you go, you are forced to be clear and complete. It is an added benefit that you have a solved problem behind you that you can use to provide honest motivation for the "next problem."
This process is modular. You can write it from the beginning to the end without having to do substantial recycling and revising. You can also stop when you have "enough." Or, if one line of thought runs out of steam before you have "enough" you can just start up with another new problem. No one says that they all have to fit together.
Finally, this process lets you focus all of your attention on what you find most interesting and most informative. A paper has an aim other than to inform and entertain can easily get trapped in issues that an honest person has to see as dull and uninteresting.
I'll give you more examples of this paradigm in class. I can also work with you to put your material into this form. I think you will find the experience extremely useful. You might even start to feel sorry for those folks who never engaged any alternative to the ragged old P to P+1 model.
Here you also start with a source paper P, but now the task is to put P into the form of some interesting problems and solutions. Along the way you are sure to have some original insights, but it is fine if these insights are just expositional.
The problem with which you begin will typically be harder to state than the toy examples given above. You may have to give some preliminary definitions, but you'll have to be very careful about this. You must give the reader just the background that is needed for the problem: no more and no less.
I promise that any paper that you honestly find interesting can be profitably recast using the "Problem-Solution" model. Moreover, if you do this reframing in a careful, thoughtful way, you will be astonished at how much more deeply you understand the paper.
- Almost without trying, you will have set aside tons of fluff, and you will see that the fluff was not missed in the least.
- You are very likely see that your source paper "just had one trick" and that trick could be completely mastered once you understood it in a simple, jargon-minimized context. This mastery may have eluded you if you had not gone through this rendering process.
- If you happened to find a "two trick" or "three trick" source paper --- well that is both great and unusually lucky!
- The process will have helped you to isolate those parts of your source paper that are "purely technical" i.e. the kind of messing around for which no insight ever emerges. These bits are irksome, but sometimes they are unavoidable. You may have to accept some deus ex machina, but please be reluctant to do so.
- The real bottom line is that by following this processs will have gotten to the heart of your source paper. This is a very worthwhile thing to do before collapsing back into the P to P+1 world.
I do hope that you can commit to writing a clear and interesting report. I am sure that everyone in the class is capable of writing one that would be a delight for me to read. It takes some investment of time, but the investment will pay dividends.
The hardest part of the whole process is to have the courage to be honest with yourself about what really is clear and interesting.
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