## Math Essay Format

**Choose a topic**

For instance, pick a geometry we have studied which you find interesting. Is there some aspect of it which was discussed briefly in class but which we didn't pursue? Is there some way of changing the rules which intrigues you?

If you're not having much luck, browse through the textbooks. Look at the books on reserve at the library. Talk to me.

Once you have tentatively chosen a topic, write a few sentences explaining it. If you are creating your own model, describe exactly what it is. If there's something missing from a proof, or from the coverage of a topic in one of the books, or whatever, describe what's missing.

Turn in your choice of topic, together with the brief explanation.

**Make an outline**

Now that you have chosen the topic, you should know at least in principle what geometry model(s) you will be working with. The next step is to decide what questions to ask about it. So make up a list of questions about your model. Does it need a distance function? Do you plan to determine what corresponds to circles?

Select several of these questions (1 is too few; 10 is too many) which you hope to answer while writing your essay. Divide them into appropriate categories. Now you're ready for the outline: Start with an introduction, end with a summary/discussion/conclusion, and put the various (categories of) problems in the middle. Briefly describe each part.

Turn in your outline.

**Do the math**

Solve the problems. This is the fun part!

**Make a rough draft**

Write up what you did. You need to include enough detail so that people can understand it. Most calculations should be given explicitly. Lots of figures (with suitable captions/descriptions) are a big help. But you also need to include enough words so that people can understand it; theorems and proofs may be appropriate, but are certainly not sufficient.

Turn in your rough draft.

**Rewrite as needed**

Be a perfectionist. Fix your math mistakes. Fix your grammar mistakes. Fix your spelling mistakes. Make sure your logic is sound. Make sure your reader will know at each stage what you're doing. Perhaps some reminders are needed: ``*Now we will solve the Dray conjecture*'' or ``*We therefore see that the Dray conjecture is false*''.

**You must do some math**

A discussion of the history of non-Euclidean geometry is not appropriate. A comparison of different (historical) versions of neutral geometry might be.

**Your work must be original**

This does not necessarily mean that you must do something nobody's ever thought of before, although you'll certainly get brownie points if that is the case. You do need to work through the math yourself, and present the results in your own words.

**References must be cited**

You may use whatever references you can find which might be appropriate. But you must give appropriate credit. A direct quote, for instance the statement of a postulate or a theorem, should be clearly labeled as such. A figure which appears elsewhere must be so labeled. It is not appropriate to make minor changes in text, or to redraw a figure, without giving a proper reference; this is plagiarism. By all means paraphrase an argument you find elsewhere. But give credit to the author. And don't fill up the entire essay this way; that's a book report.

Your references should appear separately at the end of your essay, with a section heading such as **References** or **Bibliography**. Full publication data must be given, including title, author(s), publisher, and year. Page numbers may be given if appropriate.

**Readability**

Your essay should be easy to read. Ask a friend to read it. Tell them not to worry about the details. Is the argument clear? They should be able to read the introduction and conclusion and tell you what your essay is about. Can they?

Your essay should be easy to read in another sense: Type it (or use a word processor)! Get that new ribbon/cartridge you've been thinking about! Use section headings. Indent your paragraphs. Don't run lengthy equations into the text - display them neatly on separate lines. (You may hand-write equations if you can not type special symbols.)

**Figures**

By all means include lots of figures! These can appear in the text or on separate pages at the end, and may be hand-drawn. Each one should have a label such as **Figure 1** as well as a caption. You *must* describe each figure in the text in enough detail so the reader can figure out why it's there.

**Length**

Your essay should be 5-7 pages long *not* counting figures and lengthy equations. Somewhat longer is OK; shorter is not.

**Mathematical content**

It's a good idea to get the math right!

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

__Formal Exposition__

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__

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__

Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically.

__Definitions__

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
- Abstract
- Introduction
- Body of the work
- Conclusion, appendix, and references

- Publication of a math paper
- Preprint archive
- Choice of the journal, submission
- Decision
- Publication

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.

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