Researching and Writing Papers

The process of collecting, digesting and finally communicating information orally and in written format is so absolutely fundamental to gaining true knowledge, it's a wonder so little is ever done in the undergraduate curriculum. It's a skill that doesn't come naturally, but takes a lot of practice to perfect.

I will not grade as if you were experienced in this area. I know what typical math courses are like in both high school and college; you read a problem, maybe flip through the textbook a little, scribble every which way on scratch paper (or worse yet on the paper you intend to turn in), and then write the answer as ``2.'' Unfortunately, there will be little employability in that kind of activity. My object here is to get you to practice going beyond that point.

I recently read a very interesting tribute to Claude Shannon (inventor of both the abstract theories of information and cryptology). He was described as someone who was unwilling to accept learning a part of mathematics well but insisted on constantly changing and rearranging it until he felt personally he could explain it to the man on the street. That truly expresses a basic goal of self-education, to take a subject and explain it in your own language and your own terms. That is the goal here, for you to acquire valuable information, think about it in your own terms, and convey it to the rest of us.

Here is some advice on notes and research.

1.

Collect your primary source(s).

2.

First look at the structure of the source: the section headings, highlighted facts or statements, appendices.

3.

Examine the references just to see the titles of things that proved useful to the author of your source.

4.

Record in your notes the bibliographic data (author, title, publisher, year of publication, page numbers, web address, etc.). If you see an interesting reference, write that down as well.

5.

Read the introduction first, and the conclusions second. When you read a technical document, it's always more efficient to skip back and forth, rather than reading straight through.

6.

Record the basic objectives of the source in your notes. Keep those in mind as you begin to look at the discussion sections.

7.

As you read the discussion sections, write down any highlighted statements or facts, interesting historical points or dates, etc. Some people use index cards for this sort of thing. I generally use a notebook, but I am very careful about what I write down, and sometimes I end up rearranging the pages of my notes.

8.

Take time to think about what the source is saying. Thinking about technical information is always a process of ``specializing and generalizing.''

• If there is something that is confusing you, try to write down a specific question about it in your notes.
• If the source says something in general is true, try to think of an example. For example, if the source gives a method of encoding messages by some system, try a sample message and see if you can successfully encode it yourself.
• If the source gives a collection of examples, see if you can find or express a general pattern for the examples. For instance, we described the ``homophone cipher'' early in our discussions, and then when we turned to abstract cipher systems, we never discussed how the ``homophone cipher'' fits into our general theory. That would be worth thinking and writing about.
• Sample calculations: work these out on scratch paper, but be sure to record them cleanly in your notes. Sample calculations can go a long way in learning how to explain a subject.

Some advice on writing and format:

• Write in short, clear sentences, as a journalist would. Go over your sentences and eliminate needless words. Each sentence should have a subject and verb and possibly one or two supporting phrases.
• Do not use any notation that you have not fully explained in words. Do not leave anything to the reader's imagination. For example, the sentence ``Let x be an integer'' or phrase ``for all integers x'' identifies the type of object x represents.
• Think about breaking your writing in paragraphs. Basically, a paragraph is a discussion of one idea, usually introduced in the first sentence. Sections should consist of paragraphs described a collection of related ideas.
• Be as precise as possible in your writing. Statements such as ``This will work'' leave it up to the reader to decide what ``This'' means. Explicitly state ``This program ...'' or ``This method ...'' or whatever else may be appropriate.
• It is usually clearer to organize technical arguments or calculations or data in ``blocks'' that are set off from the surrounding paragraphs. This includes large equations such as
  
  $$I = \frac{\left( \sum_{i=1}^{26} n_i^2\right) -n}{n^2-n}$$
  
  
  If you have difficulty with equations in your word-processor, and you need them, it usually appears nicer to leave space for the equation and ink it in carefully in the final paper.
• Number your pages and your sections, so that you may cross-reference parts of your paper by these numbers. If you refer to certain facts or equations in your discussion, you should name or number these facts or equations and refer to them by that name or number.

Papers on historical or literary subjects will probably have less technical information in them. However, such a paper should try to establish relations between the subject and cryptological principles and techniques. Some technical discussion may be necessary for that purpose.