Volume 4, Issue 24

August 7, 1996

Students are then encouraged to look for extensions or generalizations of material covered in class. For example, the Algebra text by Hungerford defines the greatest common divisor of a set of ring elements. A natural question that arises is whether one could similarly define the least common multiple. This is not covered in that text and so makes a good project question. During the first few class periods, I will stop class at spots in the material where I think a good project question could be asked. Many times students will come up with questions that I had not anticipated.

Once a student has a project question, they use the MathSci disc to find articles related to that question. In any of my graduate or upper level undergraduate courses, students are required to hand in research paper summaries. Guidelines for reading a research paper and writing a research paper summary are given in Hobbs' essay. I recommend summarizing articles that are 2-10 pages in length and whose title and/or abstract indicate that the student may understand some of what is written in the article. Although this is not always possible, I encourage students to summarize articles related to their project question. In the project report that students hand in at the end of the quarter, they write an account of their search. This report includes what words and phrases were searched and the number of entries found at each stage. They also hand in a printed copy of the abstracts of the (no more than 10) articles most closely related to their question. Finally, each student writes a summary indicating whether the question had previously been answered and if not whether they believe that it is a good question for further research.

Some of the purposes of this project are as follows: (1) Teach students to think beyond material presented in the classroom and text and to generate questions and/or problems. Some of the problems generated may be appropriate for further projects. (2) Teach students how to use the MathSci disc to find articles related to a particular problem or question.

(3) Familiarize students with the process of obtaining articles through the interlibrary loan program.

(4) Introduce students to the methods of reading/summarizing research papers (see related article "Reading Research Papers" by Arthur Hobbs).

I have used variations of this project in three different courses: Algebra, Combinatorics, and Graph Theory. In general, student response is favorable. Graduate students, especially those who intend to pursue a Ph.D., appear to appreciate the project more than Undergraduates. I believe that the skills acquired while completing such a project can be useful to any student. I also enjoy having an active classroom where students ask questions.

Anita C. Burris

Youngstown State University

anita@math.ysu.edu

Texas A&M University, College Station, TX 77843

(copyright 1993, 1996)

[Notice is hereby given that this essay may be reproduced and distributed to students freely, subject only to the condition that the essay should not be revised in any way. An AMSTeX version can be obtained directly from the author at hobbs@math.tamu.edu. His home page URL is http://www.math.tamu.edu/~arthur.hobbs/ and the AMSTeX version of the paper is available from there too.]

"...no man has the right to be ignorant." Louis L'Amour, "Sackett"

Have you ever been talking to a friend in your office about some mathematics paper you read once, years ago, and been frustrated by your inability to recall where and when? Have you ever regretted losing the overall view you once had of the papers in a subject? I used to have the same problems. Let me tell you how I solved them.

But first, we need to look at the reasons papers get read. From that, we can see where improvement is possible.

There are as many ways to read a mathematics paper as there are reasons for reading it. Nevertheless, these ways fall into broad classes, depending on the depth of understanding we must attain when we do the reading. These are:

(1) A quick scan, or overview, to get a feel for the contents and purpose of the paper.

(2) A bit deeper reading, intended to allow us to understand the overall structure of the paper and the most important points it contains. (3) A thorough reading, enabling us to use the results with confidence in their correctness.

(4) A painstaking reading, meticulously checking each step and verifying the accuracy of the whole paper.

When I was a graduate student, I did as many graduate students do: I read each paper as though it were vital to my livelihood. Painstaking reading (class 4) was my only mode of reading. As a result, I read fewer papers than I should have read, I missed the context of some of the papers, and in spite of the intense reading I have forgotten much more than I recall of most of the papers.

Now, except when I just want to know what the paper is about (class 1), I always begin with a class 2 reading. Even when I later complete my reading of the paper at the class 3 or class 4 level, I do the class 2 reading first. It gives me a mental structure to which to tie the individual results, and it guides and speeds the later study of the paper.

Since you cannot make sense out of theorems written using words and symbols you do not know, a class 2 reading requires mastering the most important terms and functions defined in the paper. Also, to be of use five or ten years later, the reading must generate something of archival value. What I do, then, is to write a summary sheet while I am reading.

I start by writing the identification of the paper (authors' names (the first person's name underlined in red), title, journal, date, and page numbers) on a sheet of 3-ring notebook paper. As illustrated in the example at the end of this note, I place with the identification a statement of why I am reading the paper. (For example, I write "refereeing for journal X," "Math Reviews," "research," "friend's paper," or "preparing for a visit with colleague Y.") I also give the date of reading and either a statement that I have a copy of the paper or the library call number where I can find the paper. I place this information at the bottom of the sheet, as shown in the example.

Advantages of this structure: A lifetime of reading papers can easily result in many notebooks of summaries. Even sorted by topic, the pile within one topic is often large. It is easier to sit back in a chair and flip through the bottoms of pages than the tops. Thus uniformly placing the identifying material at the bottom of the page speeds review of the reading.

As for the details of the identification, I can judge the depth of reading that I did years ago by the reason that I wrote for doing it. The date helps in a similar way: As my understanding of a subject grows, I may decide I need to reread papers read long ago to improve my understanding of their nuances. It helps to know which papers those are. In addition, the date is useful when I am doing an extended review of literature over a short period of time since, as my understanding increases, I can identify papers for later rereading which I read early and with perhaps an inadequate knowledge of the subject. Finally, knowing where to find the paper again is obviously useful.

I used to make marginal notes as I read a paper, often just trying to make it easy for me to relocate fundamental but complicated definitions or theorems. Even with these notes, often finding a definition on page 2 (or was it page 4?) that I needed on page 10 was time-consuming. Worse, I accumulated two filing cabinets full of papers with marginal notes, none of which is the slightest use to me unless I have the paper in my hand.

I still make those marginal notes, albeit more thoroughly, but now they are on my summary sheets. Beginning at the top of the sheet, here are the definitions for which I used to diligently search in the margins (see the Example). Now they are all on one sheet of paper.* Moreover, I remember them better because I wrote them out on the summary sheet. This speeds the reading of the paper and increases my comprehension of its contents.

[* In the example, it will be seen that quotes of parts of the paper might be included. Copyright is not an issue because the summary is for private use only, not for publication, and plagiarism is not possible since the source is clearly indicated.]

Here also are the statements of the key theorems, gathered in one place for my use. The value of this is obvious, but it is increased by the commentary I frequently place before, around, or after the theorems. Since these notes are for myself only, I can be as stupid as I wish in these notes, even making outrageous blunders in my commentary. That's okay. I often catch the blunders on rereading and make new commentary on them. The presence of such errors helps me see where others may go astray when I am writing up my own work. I write somewhat better as a result.

I am sure you have noticed that much of what we mathematicians write consists of lots of details used to make a few straightforward ideas intelligible and believable to the reader. Many proofs consist of one or two ideas and a lot of supporting detail. I identify these ideas by the following criterion: If I know the ideas, I can use routine mathematical techniques and standard results and methods from textbooks to complete the proof. Having noticed one or more ideas in a paper, if any of these ideas seems particularly interesting, I will write it into my summary. It helps me remember the idea and maybe use it in my own work.

"Humph!" you say. "Only a genius could routinely find the key ideas in a quick reading of a difficult paper." Of course you are right about the difficulty of catching the ideas. Fundamental ideas may or may not be highlighted by the author of a paper. Indeed, some papers seem to be written by authors who have no clear idea of what their main idea is, and some others are written by authors who write so badly I cannot tell what the idea is even when they say they are saying it. I have even suspected that some authors are trying to conceal their ideas while still publishing their results. But sometimes it is possible to recognize a good idea, and if it is new to me it goes into my summary. A decade's worth of that sort of thing can result in a lot of ideas being collected.

Naturally my summary also includes any new and interesting examples and figures. Indeed, I have seen papers in which the examples are more important than the results reported. Because examples can impose limitations on conjectures and can suggest ideas in areas remote from the problems that generated the examples, I also keep a separate 3-ring notebook of interesting and useful examples. I add to this notebook while I am making summaries of papers.

I now have several 3-ring notebooks of these summaries, accumulated over the past 9 years. First separating the summaries by research topic, and marking each section by a labeled index tab page, within each topic I store the summaries alphabetically by first author. If a summary has broad application to several different research topics, I duplicate it and store it under each of those headings.

I am now seeing the benefit of the colleciton. It is easy to review my reading and so get up to speed on a research project I had laid aside. It is useful to have an easily accesssible fund of ideas when I am doing research or preparing an advanced class. When a friend drops by to discuss research, we may touch on something I read about several years ago. "I vaguely recall seeing something on that a couple of years ago," I say. "I think Joe Smith wrote the article. Let me look." I pull out the volume containing that topic and look up Joe Smith. Scanning through the one-page summaries of his half-dozen papers there, within a minute I find the one I thought I recalled. Thus our conversation progresses. Sometimes a reference found through a summary even solves our problem.

EXAMPLE

Def: Geometry of position: That "part of geometry [that] is concerned

precisely with that which can be determined by position alone, and with the investigation of properties of position." Euler chose to investigate the Koenigsberg bridges problem, seeing it as

an opportunity to begin the study of the geometry of position. Note: This is the first paper in graph theory. Structure of paper:

1. State problem (with the famous drawing). 2. Generalize problem - arbitrary number of bridges (lower case

letters) and land masses (upper case letters). Note: New idea: Represent land masses by single points.

3. A bridge crossing is represented by a pair of upper case

letters describing the transition across a bridge from one land mass to the other.

4. Translate problem to problem of generating a sequence of

upper case letters. If there are k bridges, the sequence includes k+1 letters.

5. Count the number of occurrences of a letter. If land mass A

is the end of an odd number r of bridges, A must appear (r+1)/2 times in the sequence (Part 8), while if A is the end of an even number r of bridges, A must appear r/2 times in the sequence if the sequence does not start at A and (r+2)/2 times if the sequence does start at A (Part 11). 6. By counting, Euler shows the Koenigsberg route is impossible

(Part 9 of paper).

7. The famous theorem, that an Eulerian circuit requires even

degree at every vertex and an open Eulerian trail requires even degree at every vertex but two, is hinted in Parts 11-13, necessity is shown in Parts 16-17, and the theorem is stated in Part 20.

8. Examples and tabular description and solution of examples in

Parts 14-15.

9. Method of finding a circuit or trail: suppress pairs of

multiple edges and easily find trail in result (Part 21). Overall: Euler shows the necessity of his theorem, but his attempt to

show sufficiency of the condition (by reference to his tabular method) is not convincing.

L. Euler, Solutio problematis ad geometriam situs pertinentis, Commentarii

Academiae Scientiarum Petropolitance 8 (1736), 1741, pp. 128-140. (Reprinted and translated in H. Fleischner, Eulerian Graphs and Related

Topics, Part 1, Vol. 1, North-Holland, Amsterdam, 1990, pp. II.2-II.19.)

example have

Jan, 1993

EXAMPLE