*Concerns of Young Mathematicians*
Volume 4, Issue 24
August 7, 1996
An electronically distributed digest for discussions
of the issues of concern to mathematicians at the
beginning of their careers.
Studying a Research Question
The following is a description of a project that I give in graduate
and upper level undergraduate courses. On the first day of class, students
are told that some time within the first half of the quarter (this allows
time for the interlibrary loan process if necessary) they must each find
a question or problem not covered in the text. They are given a copy of
the essay "Reading Research Papers" by Arthur Hobbs and a handout explaining
how to use the MathSci disc. The handout leads them step by step through
a sample search and gives a few tips on choosing words or phrases to search.
The sample search question is one that could be asked about material covered
on the first day of class.
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
Reading Research Papers
by Prof. Arthur M. Hobbs
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 firstname.lastname@example.org. 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
(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
(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
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
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.
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
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
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.)