Writing about mathematics is a unique challenge for students. While all disciplines share some of the same rules of good writing,
math papers requires students to tackle both conceptual and grammatical challenges they won't encounter anywhere else.
Here are some of the things you need to know if you're going to be writing about mathematics.
Voice: active
Style guide: Manual for Authors of Mathematical Papers (from the American Mathematical Society)
Person: 1^{st} or 3^{rd}
Central argument: research question
Organization: introduction, body, and conclusion
Note: Different style guides will handle the issues presented here slightly differently, particularly when it comes to punctuating and aligning mathematical equations. This article will handle general rules, but if you have specific questions you should consult whichever style guide your teacher has requested you use.
Types of math papers
Most student math papers will be a combination of computations and theoretical explanation. For example, you may be asked to derive a formula that describes a certain population, then use that formula to predict future growth or decay. More advanced students will likely be writing papers that address purely theoretical questions or proofs of theorems.
How to write about math
Tell, don't just show
Students often think that when
writing a math paper all they have to do is show their work, i.e., plop down the equations they used, then show the results. But writing about math is about more than just arriving at the right answer; it's also about showing how you got there. Professional mathematicians painstakingly document every step they take in deriving and applying equations, and student writers need to do the same. So, don't just put a formula in your paper and except that to be enough. You need to tell the reader where that formula came from, why you choose that particular formula, what all the parts mean, and how it fits in the larger scope of your work.
Organization
The general structure of a math paper will be a lot like any other
essay. You'll have an introduction, body, and conclusion (your paper will also likely need sections like an abstract and references as well).
- Introduction: Tell the reader the purpose of your work and explain why it's important. You may also need to give background by citing related papers. Often in math papers the introduction will include a preview of the solution or argument you're about to make.
- Body: Unlike a research paper, there's no set style for the body of a math paper. Instead, each will be organized according to the particular assignment. This might mean breaking it down into subheadings that each answer a particular question or, for shorter papers, you might just have one big body section. Whichever you choose, the key to good organization is for the paper to logically flow from one idea to the next.
- Conclusion: Wrap up your paper by summarizing what you've found and discussing avenues for further research.
Formal and informal structure
Math papers consist of two main parts working side by side. The formal, or logical, structure of your paper is the definitions and proofs you are using to build your argument. The informal material is any additional text such as examples, analogies, and explanations. Ideally, these two element should be woven together to create a paper that has a solid logical foundation upon which explanatory material is added to help the reader.
Defining terms
Defining terms isn't just a side job-it's an integral part of your argument. How you define terms can determine whether your argument makes sense, and defending those definitions is often part of the
paper writing process. Even if you're not being controversial, you still need to clearly and explicitly define every variable and symbol you use before your use them, and make sure that they are used consistently throughout your work.
Know your vocabulary
Just like every field, math has its own terminology, and there are many words that have a meaning specific to the field of mathematics. For example, you should know that the term theory is applied sparingly only to major results, while the terms proposition and corollary refer to minor results derived from other theories, propositions, or corollaries. A lemma is a statement used in a proof. Using these terms incorrectly will call the quality of your work into question and likely lower your grade.
The look of the page
When writing, pay attention to how your text looks on the page. Most mathematical papers are a mix of equations and text, and any paper that has too much of one or the other is likely going to be hard to read.
Style
Voice and person
In general you should use the active voice whenever possible. Unlike in science writing, its acceptable to use the 1^{st} person (I or we).
Equations and formulas
Equations and formulas should be set apart on separate lines, like this:
Equations should be integrated into the preceding and proceeding sentences, meaning they should never just be dropped onto the page without any sort of explanation.
Equations should also be treated as a part of the sentence. You might have noticed in the example above that the equation ends with a period. That's because it ends the sentence that came before it. When equations don't end the sentence, they don't require punctuation. For example: We use the equation for binomial expansion
to find the coefficient for certain terms (that equation does not require a period). If you're not sure, try reading your work out loud. The formulas and equations should flow seamlessly with the rest of your work.
What to avoid
Lists of formulas
Math papers are not about showing that you've done the calculations, but instead about demonstrating that you understand the ideas behind the formulas. When you just list equations or figures without any corresponding commentary, you aren't going to be showing that you understand the material. Instead, you need to explain what those equations mean, where they came from, and why they're important. If you have a big list of formulas without any commentary breaking it up, then you're doing something wrong.
Inconsistency
Mathematical paper require the use of many symbols and abbreviations, and it's important that these be consistent throughout your work. If you assign a particular variable to a value, make sure that you use that variable the whole time; you also want to be sure that each variable is used only once (i.e., don't assign the same variable to two different values). Also be sure to write equations consistently. If you write "x - 1 = 0, therefore x = 1," don't later write "x - 2 = 0 -> x = 2." In other words, use the same words or symbols that you used earlier to do the same job.
Relying on symbols
It might seem like a good idea to use symbols instead of words. After all, math is all about equations and formulas. But a math paper is just like in other paper in that it's not acceptable to use abbreviations and symbols instead of their corresponding words. So, in the example in the paragraph above, it would be incorrect to use the -> symbol. Instead, you'd say "therefore" or "demonstrating that."
Bad formatting
Formatting all those equations and formulas can be tricky, which means you need to get comfortable with whatever word processor you're using. Learn how to use the equation editor in Word or how to use the various
math writing software packages so that your paper is consistently formatted and easy to read. If your equations are all over the page or your variables are different sizes, your work will look unprofessional.