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A research paper is primarily a discussion or argument based on a thesis, which

includes evidence from several collected sources.
While it may seem like a monumental project, it is really a straightforward process that
you can follow, step by step. Before you get started, make sure you have plenty of note
paper, multi-colored highlighters, and a pack of multi-colored index cards.
You should also read over the checklist for research ethics.
Organizing your Research Paper
You'll use the following steps to complete your assignment.
1. Select a topic
2. Find sources
3. Take notes on colored index cards
4. Arrange your notes by topic
5. Write an outline
6. Write a first draft
7. Revise and re-write
8. Proofread
Library Research
Find a comfortable place where you wont be distracted by people passing by. Find a
table that provides lots of space, so you can sort through several potential sources, if
necessary.
Become familiar with the services and layout of the library. There will be a card catalog
and computers for database searches.
Select a Research Paper Topic
If you're free to choose your topic, find something that youve always wanted to know
more about. If you have a fascination with weather or you watch every TV show you can
find on tornadoes, for instance, you may want to find a topic related to that interest.
Once you narrow your choices to a specific subject area, find three specific questions to
answer about your topic. A common mistake by students is to choose a final topic that is
too general. Try to be specific: What is tornado alley? Are certain states really more
likely to suffer from tornadoes? Why?
One of your questions will turn into a thesis statement, after you do a little preliminary
research to find theories to answer to your questions. Remember, a thesis is a
statement, not a question.
Find Sources
Use the card catalog or computer database in the library to locate books. (See Sources to
Avoid.) Find several books that seem to be relevant to your topic.
There will also be a periodical guide in the library. Periodicals are publications issued on
a regular basis, like magazines, journals, and newspapers. Use a search engine to find a
list of articles relating to your topic. Make sure to find articles in periodicals that are
located in your library. (See How to Find an Article.)
Sit at your work table and scan through your sources. Some titles can be misleading, so
youll have some sources that dont pan out. You can do a quick read over the materials
to determine which ones contain useful information.
Taking Notes
As you scan your sources, you will begin to zero in on a thesis. Several sub-topics will
also begin to emerge. Using our tornado topic as an example, a sub-topic would be the
Fujita Tornado Scale.
Start taking notes from your sources, using color coding for the sub-topics. For instance,
all information referring to the Fujita Scale would go on orange note cards.
You may find it necessary to photocopy articles or encyclopedia entries so you can take
them home. If you do this, use the highlighters to mark the useful passages in the
relevant colors.
Every time you take a note, be sure to write down all bibliographical
information to include author, book title, article title, page numbers, volume
number, publisher name and dates. Write this information on each and every
index card and photocopy. This is absolutely critical!
Arrange Your Notes by Topics
Once you have taken color-coded notes, you will be able to sort your notes more easily.
Sort the cards by colors. Then, arrange by relevance. These will become your
paragraphs. You may have several paragraphs for each sub-topic.
Outline Your Research Paper
Write an outline, according to your sorted cards. You may find that some of the cards fit
better with different colors or sub-topics, so simply re-arrange your cards. Thats a
normal part of the process. Your paper is taking shape and becoming a logical argument
or position statement.
Write a First Draft
Develop a strong thesis statement and introductory paragraph. Follow through with your
sub-topics. You may find that you dont have enough material, and you may need to
supplement your paper with additional research.
Your paper may not flow very well on the first try. (This is why we have first drafts!)
Read it over and re-arrange paragraphs, add paragraphs, and omit information that
doesnt seem to belong. Keep editing and re-writing until youre happy.
Create a bibliography from your note cards. (See citation makers.)
Proofread
When you think you are happy with your paper, proof read! Make sure it is free of
spelling, grammatical, or typographical errors. Also, check to make sure youve included
every source in your bibliography.
Finally, check the original instructions from your teacher to make sure you are following
all assigned preferences, like title page directions and placement of page numbers.

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by Professor Shahn Majid
The style and format of research papers varies from subject to subject (and indeed journal to journal).
This guide is aimed at students in the mathematical sciences.
These are some hints for starting PhD students on how to write papers. It is assumed of course that
you have some results worth presenting (as no amount of good writing can cover up a lack of
content).

How you write depends on the journal/type of reader you are addressing. Also, keep in mind some
role models -- people you know or famous papers. The general aim is to be attractive to non-experts
as much as can be expected, while interesting and not offensive to experts.
The Title
This should instantly convey why your work stands out from all previous ones. Should be intelligible to
non-experts and down-to-earth though perhaps slightly enigmatic or 'catchy'.
The Abstract
The significance of a paper tends to be in inverse proportion to the length of the abstract. The shorter
the abstract, usually the more powerful the results. So the challenge is to keep it concise while at the
same time conveying the key results and ideas behind the paper. The abstract should be self-
contained and intelligible before one has read the paper.
Keywords
This is for computer database searches to pick up on, along with words in the title and abstract. So
think about what kind of search items you would want leading to your paper. This goes along with
finding a Math Reviews code, which can be included in a foot note if you know it.
The Intro
Many readers and (sad to say) quite possibly the referee will not get past the introduction. So it should
be beautifully written with much work. Up to a quarter of the manuscript might reasonably be taken up
by the introduction and the preliminaries. Most people find it easier once they get onto the
mathematics itself. You should try in the introduction to cover the following checklist.

i) The motivation. This should recall to the reader why the kind of result mentioned already in the
abstract would be interesting and important. It also tells the reader what you think is the motivation, so
that if he or she agrees with the way you are looking at the field, there's some probability that the
paper will be useful for them. Keep it as down to earth as possible.

ii) The results and strategy -- the key behind the work. Don't just repeat the abstract. Don't be
ashamed or too proud to admit and reference the previous work ('the shoulders of giants') which
inspired and led up to your result. A good way is to tell a story, an interesting one that puts everything
into perspective re the existing literature and conveys how it is you succeeded where others failed.
What was the key idea which nobody else spotted? It should not reflect the actual historical progress
of your research (which may have been long and winding) but rather based on how your thinking
should have gone with the benefit of hindsight. This is not quite the same as the shortest logical path
(which would not be understood until after the paper is read), but rather involves an historical element
with reference to works and ideas that the reader might already be familiar with. Note that it's rare for
a young person to do something totally out of the blue, and worrisome for a referee.

iii) Survey the field so far. Make contacts with other aspects of the literature. Try to connect or
reference all the relevant players in the field. This takes knowledge of the literature and above all a
sense of historical perspective. Who did really introduce the idea X that you are using and are giving
him or her proper credit? This can also be woven into the above by way of making it interesting.

iv) Outline the organisation. This should be brief but not simply a list. State the goal and main
achievement of each section. Make it into a story whereby each section is logically a precursor to the
next section.

v) Preliminaries. This should include technical remarks on notation to be used and basic references
such as books for conventions. You can recall in this section for clarity things that you should be
ashamed to publish in the later sections. If a lot of machinery which you did not invent is to be used,
this is the place to develop it or give references. Remember, however, that you are not writing a thesis
here: your goal is not to redo the work of persons A, B, C (which you may well have done in your
notes while reading them, but that's tough). The goal is rather to make enough concise references or
explanations so that exactly how you intend things to be defined, which conventions exactly you are
using etc are all completely clear. Find ways to state cut-and-dry and precise definitions that the
reader will be able to refer back to when reading the paper, without digressions or story-telling.

Anything in this section is 'safe' in the sense that the reader does not assume that this is your work.
Indeed, the reader assumes it is not (and the referee can always make you delete it if it's too much).
(Just the opposite is true in later sections, where you should not repeat well-known results or if forced
to do so, explain that it is `for completeness ...' and give completely unambiguous references to the
literature). So try to put most of what you will need here in the preliminaries.
Sections 1--n
Finally you get to explain your new results. Each section should begin with a recall of the goal and
strategy of the section in case the reader forgot. Each section should have a main achievement.

Then proceed as clearly as possible in the correct logical order. Don't try to save space by building
into your propositions repeats of other people's results. i.e. they should contain only results that are
new, no matter how logical it would be to mention the other results not proven by you (that would be
OK in a book or thesis or review article, but research papers should only contain the incremental
data).

In other words, some of the stuff you want to put down is all part of the beautiful logical picture, but
that's too bad. Unless you personally have something new and worthwhile to say about it, you have
no business to be recalling it here (maybe in the intro with citations as motivation) and also should not
be building it in mixed with your own results. As well as the logically-dictated tendency to repeat, we
all have a human weakness to think that what we spent hours figuring out for ourselves is partly ours.
This is a demon to be resisted. Previous work is previous work and don't be too proud to say you are
using it, and whose it is you are using. You should ask yourself how would you feel if somebody
developed your work and integrated it into theirs without being clear about your contribution.

Results can be organised as lemmas -- technical results you need later but not of self-contained
interest, propositions -- moderately interesting new results, and theorem -- main new results. Each of
these should be an irreducible 'gem': i.e. break up theorems etc with disjoint parts into propositions
leading up to the denouement of your main theorem. You can follow these with some corollaries,
which are more like tasty desserts.

The proof of a theorems or proposition should be substantial and not a cheap logical trick in which it's
immediate from some other work -- that's a corollary or a remark. Ideally, the proof of the main
theorem should use as many as possible of the lemmas and propositions already proven, to show
that they were all needed and worthwhile.

Statements of theorems etc should be as self-contained as possible. Under this constraint, the shorter
ones are the most powerful, i.e. pack the most punch. A `punchy' theorem can be achieved by
properly setting up the relevant background in the preamble and keeping background material out of
the statement itself (as much as possible that is consistent with being self-contained notationally). The
statement itself should be boiled down to the part that is really new and important.

The end of the section is a good place to put any informal remarks. Anything you want to claim, assert
or conjecture but which you haven't thought through formally to make a theorem, can appear here.
Things are easily forgiven at the ends of sections if the section already had good results in it.

These remarks could also lead onto the next section. But don't overdo that since the beginning of the
next section is going to reintroduce itself anyway. I.e., if you're setting up the next section it should be
in a subtle way that doesn't overlap with the official set-up which will appear there.
General Guide to Style
Bad writing often goes hand-in-hand with murky thinking, so by writing clearly you are forced to clarify
your understanding also. Thinking about layout, ordering of sentences and even simple things like
punctuation are very important and can have a surprisingly good effect on your own understanding of
the material.

To some extent, the best rule of good writing is to write and write. Eventually it gets better. In the
meantime, some things to watch out for are as follows.

Avoid non-sequiturs. Sentences should logically lead on from one to the next as smoothly as
walking. English has a preference for short sentences with a great deal of structure connecting across
sentences. Words or ideas used a few sentences back will still be in the reader's mind, so there
should not be any jarring change of topic. If you want to change the topic, no problem, but warn the
reader by key phrases like 'on the other hand', 'meanwhile', 'in contrast to this', 'moreover' etc.

A shift of general topic is signalled by a new paragraph. Again, previous paragraphs are still active in
the readers mind so any very big shift should be excused by a suitable explanation like 'Now we come
to ...' or 'To conclude this section' or other orientation signal. The signals could refer back to the
introduction and outline, or might indicate a surprise for the reader.

Avoid making sandwiches. A conceptual sandwich is where you begin with one idea, move on to
another, and then move back to the first one. This can happen at all scales: within a paragraph, within
a section or in the overall layout of the paper. It indicates poor organisation and should be avoided.
Can you move the middle of the sandwich to the top or the bottom, thereby pooling together the two
related topic? The more general topic should usually come first, with the more specific sub-topic
following, unless you deliberately want to be pedagogical. The idea of avoiding a sandwich is that
when you bring up a topic, say all that you will want to say about it in the near future, before moving
on to further questions arising from it. Chopping and changing uses up the reader's energy.

A similar phenomenon can occur with a sentence too. A common problem is that the second half of
the sentence came as an afterthought but more properly belongs as the first half of the sentence. So
always ask yourself if you should reverse the order of a sentence.

Validation status of assertions. Every assertion should have a clear validation status. By this I
mean that it should be clear to the reader from context or from signals in the syntax exactly how the
reader is supposed to know that the assertion is correct. Is it (i) supposed to be self-evident from what
was just said (ii) supposed to follow from something said a while back (give a signal to where) (iii) a
well-known fact that the reader should know anyway (iv) a fact proven elsewhere by somebody else
which you had cited a while back (cite them again `on the dot' if there is any possibility of ambiguity).
(v) a fact that you will be justify later (give the forward reference), etc ?

Run-on sentences. Some languages have long sentences with lots of commas, but English does not
have the grammar to support this. Rather, sentences should be short and sharp. Russians say that
English people sound like barking dogs. A common fear is to avoid losing the context by finishing the
sentence, leading the author to put a comma and run on with another one. There is no need to be
afraid of that because words will still remain active for a short while after the period. A good rule is to
look for sentences longer than one or two lines and see if ', which' or ', where' etc can be replaced by
fresh sentences.

'This' and 'it'. Beware of pronouns like 'this' and 'it'. Is it absolutely clear and unambiguous what they
refer back to? You may know what you had in mind but will the reader? And don't use 'this' for 'the
present'.

'Never' and 'only'. Beware of 'never' and 'only'. These are strong assertions and unless you've really
proven them it's best to water them down with 'appears to be' or 'it seems'. However, don't use
'probably'.

Commas. Commas are especially important but hard to give rules for. Don't go by where you pause
when speaking; at best commas can be used as a kind of 'conceptually pause' or to make an aside
with the help of a later comma (brackets are usually better, however). Very often a fresh sentence
would be better. And when writing your thesis or for camera-ready work you should be more polished
about punctuation rules: consult a style manual like Fowlers.

Math symbols. Try not to begin or end a sentence with a math symbol. More precisely, try to avoid
math symbols clashing with textual punctuation. The exception to this is displayed equations, where
the general rule is to put commas and if necessary a final period, so that one can read through the
displayed equation like text. On the other hand, don't over do it by putting connectives like 'and' or
other substantial text into the displayed equation. The displayed equation is half-way to a table so
should be laid out for visual clarity and without unnecessary text.

Is it defined? Check that all symbols and terminology are defined to some extent before they are
used. This can be done in a formal definition or more informally. One technique in mathematical work
is to put the term for an important concept in a different font when its usage is being specified for the
first time. This is especially important in the Preliminaries section, but applies elsewhere also. On a
smaller scale, make sure that any symbols are quietly specified so that it is clear what they denote.
Don't assume that your notation is obvious or standard, since others could have grown up with
different conventions.
Concluding Remarks
A good intro and well-written paper does not need conclusions. But this is the place for epilogical
comments that can be understood only now after your new results. They are like corollaries or
informal results or consequences that you haven't worked out yet in detail. You can tell your ideas
about these if you want in the form of expected directions for further work.
Bibliography
Do a computer search (Bids, hepth, q-alg) to make sure you have picked up all relevant recent work.
Also, did you fairly reference the originators of all the works and ideas that you used? Don't go by
where you first read something (which might have been only pedagogical) but by where it was really
discovered. That takes a bit of detective work but we all have to be our own policepersons.

Good Luck and don't forget to spellcheck and check punctuation if you need it! If you can, let it sit on
the shelf for a week or a month before giving it a final fresh reading. There may well be typos that you
did not spot first time round.