Mathematics research papers are different from standard academic research papers in important ways, but not so different that they require an entirely separate set of guidelines. Mathematical papers rely heavily on logic and a specific type of language, including symbols and regimented notation. There are two basic structures of mathematical research papers: formal and informal exposition.
Structure and Style
The author must start with an outline that develops the logical structure of the paper. Each hypothesis and deduction should flow in an orderly and linear fashion using formal definitions and notation. The author should not repeat a proof or substitute words or phrases that differ from the definitions already established within the paper. The theorem-proof format, definitions, and logic fall under this style.
Informal exposition complements the formal exposition by providing the reasoning behind the theorems and proofs. Figures, proofs, equations, and mathematical sentences do not necessarily speak for themselves within a mathematics research paper. Authors will need to demonstrate why their hypotheses and deductions are valid and how they came to prove this. Analogies and examples fall under this style.
Conventions of Mathematics
Clarity is essential for writing an effective mathematics research paper. This means adhering to strong rules of logic, clear definitions, theorems and equations that are physically set apart from the surrounding text, and using math symbols and notation following the conventions of mathematical language. Each area incorporates detailed guidelines to assist the authors.
Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically.
In order for the reader to understand the author’s work, definitions for terms and notations used throughout the paper must be set at the beginning of the paper. It is more effective to include this within the Introduction section of the paper rather than having a stand-alone section of definitions.
Theorems and Equations
Theorems and equations should be physically separated from the surrounding text. They will be used as reference points throughout, so they should have a well-defined beginning and end.
Math Symbols and Notations
Math symbols and notations are standardized within the mathematics literature. Deviation from these standards will cause confusion amongst readers. Therefore, the author should adhere to the guidelines for equations, units, and mathematical notation, available from various resources.
Protocols for mathematics writing get very specific – fonts, punctuation, examples, footnotes, sentences, paragraphs, and the title, all have detailed constraints and conventions applied to their usage. The American Mathematical Society is a good resource for additional guidelines.
LaTeX and Wolfram
Mathematical sentences contain equations, figures, and notations that are difficult to typeset using a typical word-processing program. Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing.
LaTeX is highly recommended for researchers whose papers constitute mathematical figures and notation. It produces professional-looking documents and authentically represents mathematical language.
Wolfram Language & System Documentation Center’s Mathematica has sophisticated and convenient mathematical typesetting technology that produces professional-looking documents.
The main differences between the two systems are due to cost and accessibility. LaTeX is freely available, whereas Wolfram is not. In addition, any updates in Mathematica will come with an additional charge. LaTeX is an open-source system, but Mathematica is closed-source.
Good Writing and Logical Constructions
Regardless of the document preparation system selected, publication of a mathematics paper is similar to the publication of any academic research in that it requires good writing. Authors must apply a strict, logical construct when writing a mathematics research paper.
There are resources that provide very specific guidelines related to following sections to write and publish a mathematics research paper.
- Concept of a math paper
- Title, acknowledgment, and list of authors
- Body of the work
- Conclusion, appendix, and references
- Publication of a math paper
- Preprint archive
- Choice of the journal, submission
The critical elements of a mathematics research paper are good writing and a logical construct that allows the reader to follow a clear path to the author’s conclusions.
That is the theory, at least. But how do you go about doing it all? This brief guide provides some answers.
Teaching YourselfFrom the outset, keep in mind one important point: Writing a research paper is in part about learning how to teach yourself. Long after you leave college, you will continue learning about the world and its vast complexities. There is no better way to hone the skills of life-long learning than by writing individual research papers. The process forces you to ask good questions, find the sources to answer them, present your answers to an audience, and defend your answers against detractors. Those are skills that you will use in any profession you might eventually pursue.
The Five Commandments of Writing Research PapersTo write first-rate research papers, follow the following simple ruleswell, simple to repeat, but too often ignored by most undergraduates.
1. Thou shalt do some background reading, think hard, and speak with the professor in order to identify a topic.At the beginning of a course, you will probably not know enough about the major scholarly topics that are of most importance in the field, the topics that are most well-covered in the secondary literature or the topics that have already had the life beaten out of them by successive generations of writers. You should begin by doing some general reading in the field. If nothing else, begin with the Encyclopaedia Britannica, a wonderful but sadly neglected resource. Read a few books or articles on topics you find of interest. Follow up the suggested reading on the course syllabus or the footnotes or bibliographies of the texts you are reading for the course. After that, speak with the professor about some of your general ideas and the possible research directions you are thinking about pursuing. And you should do all this as early in the course as possible.
2. Thou shalt have a clear research question.A research question, at least in the social sciences, begins with the word why or how. Think of it as a puzzle: Why did a particular political or social event turn out as it did and not some other way? Why does a particular pattern exist in social life? Why does a specific aspect of politics work as it does? How has a social or political phenomenon changed from one period to another? The question can be general or particular. Why have some countries been more successful in the transition from Communism than others? Why did the Labour Party win the last British general election? How have conceptions of race changed in the US since the 1960s? How do different electoral systems affect the behavior of political parties?
The point is that you should attempt to identify either:
- novel trends, developments or outcomes in social life that are not readily apparent (the how questions), or
- the causes of a particular event or general trend (the why questions).
How and why questions are essential because they require the author to make an argument. Research questions that do not require an argument are just bad questions. For example, a paper on What happened during the Mexican revolution? requires the author to do no more than list facts and datesa good encyclopedia entry, maybe, but not a good research paper. What and when questions are only the starting point for writing research papers. Obviously, you need to have a firm grasp of the facts of the case, but you must then move on to answer a serious and important why or how question in the paper itself.
3. Thou shalt do real research.Real research means something other than reading secondary sources in English or pulling information off the Internet. Real research means using primary sources. What counts as a primary source, though, depends on what kind of question you are trying to answer.
Say you want to write a paper on the causes of Communisms demise in eastern Europe. You would begin by reading some general secondary sources on the collapse of Communism, from which you might surmise that two factors were predominant: economic problems of Communist central planning and Mikhail Gorbachevs reforms in the Soviet Union. Primary sources in this case might include economic statistics, memoirs of politicians from the period or reportage in east European newspapers (available in English or other languages). Bring all your skills to bear on the topic. Use works in foreign languages. Use software packages to analyze statistical data.
Or say you want to write about how conceptions of national identity have changed in Britain since the 1980s. In this case, you might examine the speeches of British political leaders, editorials in major British newspapers, and voting support for the Scottish National Party or other regional parties. You might also arrange an interview with an expert in the field: a noted scholar, a British government representative, a prominent journalist.
The point about primary sources is that they take you as close as possible to where the action isthe real, on-the-ground, rubber-meets-the-road facts from which you will construct your interpretive argument. There are, however, gradations of primary evidence. The best sources are those in original languages that are linked to persons directly involved in the event or development that you are researching. Next are the same sources translated into other languages. Then come sources that are studies of or otherwise refer to direct experience. In your research, you should endeavor to get as close as possible to the events or phenomena you are studying. But, of course, no one can speak every language and interview every participant in a political or social event. Part of being a creative scholar is figuring out how to assemble enough evidence using the skills and resources that you possess in order to make a clear and sustainable argument based on powerful and credible sources.
One other note for Georgetown students: In a city that contains one of the worlds great research libraries, representations from nearly every country on the planet, the headquarters of countless international organizations, numerous research institutes, and scores of other political, economic, cultural, and non-governmental associations and institutions, both domestic and international, there is absolutely no excuse for the complaint that I cant find anything on my topic in Lauinger.
4. Thou shalt make an argument.Unfortunately, many undergraduate research papers are really no more than glorified book reports. You know the drill: Check out ten books (in English) from the library, skim through three of them, note down a few facts or mark some pages, combine the information in your own words, and there you have it.
This will not do. Your paper must not only assemble evidencefacts about the worldbut it must weave together these facts so that they form an argument that answers the research question. There are no once-and-for-all answers in any scholarly field, but there are better and worse arguments. The better ones have powerful evidence based on reliable sources, are ordered and logical in the presentation of evidence, and reach a clear and focused conclusion that answers the question posed at the beginning of the paper. In addition, good arguments also consider competing claims: What other counter-arguments have been put forward (or could be put forward) to counter your points? How would you respond to them? In fact, consideration of counter-arguments is often a good way to begin your paper. How have scholars normally accounted for a particular event or trend? What are the weaknesses of their accounts? What evidence might be marshaled to suggest an alternative explanation? How does your account differ from the conventional wisdom?
5. Thou shalt write well.Writing well means presenting your argument and evidence in a clear, logical, and creative way. An interesting argument cloaked in impenetrable prose is of no use to anyone. Sources must be accurately and adequately cited in footnotes, endnotes or in-text notes using a recognized citation style. The writing style must be formal and serious. Tables, graphs or other illustrations should be included if they support your overall thesis.
These are only a few guidelines on how to write research papers. You will no doubt develop your own styles, rules, and techniques for doing research, making arguments, and presenting the results of your work. But if you follow the commandments above, you will be well on your way to writing good research papersand hopefully learn something about an important political or social topic along the way.
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