A Guide to Writing in Mathematics
Classes 
For most of your life so far, the only kind of writing you've done in math classes has been on homeworks and tests, and for most of your life you've explained your work to people that know more mathematics than you do (that is, to your teachers). But soon, this will change.
Now that you are taking Calculus, you know far more mathematics than the average American has ever learned  indeed, you know more mathematics than most college graduates remember. With each additional mathematics course you take, you further distance yourself from the average person on the street. You may feel like the mathematics you can do is simple and obvious (doesn't everybody know what a function is?), but you can be sure that other people find it bewilderingly complex. It becomes increasingly important, therefore, that you can explain what you're doing to others that might be interested: your parents, your boss, the media.
Nor are mathematics and writing farremoved from one another. Professional mathematicians spend most of their time writing: communicating with colleagues, applying for grants, publishing papers, writing memos and syllabi. Writing well is extremely important to mathematicians, since poor writers have a hard time getting published, getting attention from the Deans, and obtaining funding. It is ironic but true that most mathematicians spend more time writing than they spend doing math.
But most of all, one of the simplest reasons for writing in a math class is that writing helps you to learn mathematics better. By explaining a difficult concept to other people, you end up explaining it to yourself.
Every year, we buy ten cases of paper at $35 each; and every year we sell them for about $1 million each. Writing well is very important to us. 
 Bill Browning, President of Applied Mathematics, Inc. 
A good mathematical essay has a fairly standard format. We tend to start solving a problem by first explaining what the problem is, often trying to convince others that it's an interesting or worthwhile problem to solve. On your homeworks, you've usually just said, "9(a)" and then plunged ahead; but in your formal writing, you'll have to take much greater pains.
After stating what the problem is, we usually then state the answer, even before we show how we got it. Sometimes we even state the answer right along with the problem. It's uncommon, although not so uncommon as to be exceptional, to read a math paper in which the answer is left for the very end. Explaining the solution and then the answer is usually reserved for cases where the solution technique is even more interesting than the answer, or when the writers want to leave the readers in suspense. But if the solution is messy or boring, then it's typically best to hook the readers with the answer before they get bogged down in details.
Another difference is that when you do your homework, it is important to show exactly how you got your answer. However, when you write to a nonmathematician, sometimes it's better to show why your answer works, with just a brief explanation as to how you got it. For example, compare:
Homework Mathematics: and so either = 5 or = 2. More Formal Mathematics:
and also 
Math is difficult enough that the writing around it should be simple. 'Beautiful' math papers are the ones that are the easiest to read: clear explanations, uncluttered expositions on the page, wellorganized presentation. For that reason, mathematical writing is not a creative endeavor the same way that, say, poetry is: you shouldn't be spending a lot of time looking for the perfect word, but rather should be developing the most clear exposition. Unlike humanities students, mathematicians don't have to worry about overusing 'trite' phrases in mathematics. In fact, at the end of this booklet are a list of trite but useful phrases that you may want to use in your papers, either in this class or in the future.
This guide, together with your checklist, should serve as a reference while you write. If you can master these basic areas, your writing may not be spectacular, but it should be clear and easy to readwhich is the goal of mathematical writing, after all.
When you turn in your writing assignment, you should use a paper clip to attach the checklist to the front. You should feel free to use both the checklist and this booklet as a guide while you write, because you will be graded directly on the criteria outlined on the checklist. What follows here is a more detailed explanation of the criteria I use for grading your papers.
1. Clearly restate the problem to be
solved.
Do not assume that the reader knows what you're talking about. (The person you're writing to might be out on vacation, for example, or have a weak memory). You don't have to restate every detail, but you should explain enough so that someone who's never seen the assignment can read your paper and understand what's going on, without any further explanation from you. Outline the problem carefully.
2. State the answer in a complete sentence
which stands on its own.
If you can avoid variables in your answer, do so; otherwise, remind the reader what they stand for. If your answer is at the end of the paper and you've made any significant assumptions, restate them, too. Do not assume that the reader has actually read every word and remembers it all (do you?).
3. Clearly state the assumptions which underlie
the formulas.
For example, what physical assumptions do you have to make? (No friction, no air resistance? That something is lying on its side, or far away from everything else?) Sometimes things are so straightforward that there are no assumptions, but not often.
4. Provide a paragraph which explains how the
problem will be approached.
It's not polite to plunge into mathematics without first warning your reader. Carefully outline the steps you're going to take, giving some explanation of why you're taking that approach. It's nice to refer back to this paragraph once you're deep in the thick of your calculations.
5. Clearly label diagrams, tables, graphs, or other visual representations of the math (if these are indeed used).
In math, even more than in literature, a picture is worth a thousand words, especially if it's well labeled.
Label all axes, with words, if you use a graph. Give diagrams a title describing what they represent. It should be clear from the picture what any variables in the diagram should represent. The whole idea is to make everything as clear and selfexplanatory as possible.
If you decide to draw pictures in Mac Draw or Super Paint (which you can then copy to Write Now, Microsoft Word, or Mac Write ), you will (a) have a lot of fun, and (b) waste a lot of time. If you're going to draw diagrams on the computer, I recommend doing so in the early stages of the project, and then saving the document so you can change it laterbecause you'll have to change it later, and you can't do it once it's in Write Now or whatever.
(a) Even if you label your diagram (and you should), you should still explain in words what your variables are.
(b) If there's a quantity you use only a few times, see if you can get away with not assigning it a variable. As examples:
Figure 1: Diagram of the triangle 
We see that the area of the triangle will be onehalf of the product of its height and basethat is, the area of the triangle is (1/2) 3 4 = 6 square inches. 

We see that A = (1/2)h b, where A stands for the area of the triangle, b stands for the base of the triangle, and h stands for the height of the triangle, and so A = (1/2) 3 4 = 6 square inches. 
Elementary physics tells us that the velocity of a falling body is proportional to the amount of time it has already spent falling. Therefore, the longer it falls, the faster it goes. 

Elementary physics tells us that vt =g (t t ), where vt is the velocity of the falling object at time t, g is gravity, and t is the time at which the object is released. Therefore as t increases, so does vt : i.e., as time increases, so does velocity. 
I hope that you'll agree that the first example of each pair is much easier to read.
(c) The more specific you are, the better. State the units of measurement. When you can use words like "of", "from", "above", etc., do so. For example:
We get the equation d =rt, where d is the distance, r is the rate, and t is the time. 

We get the equation d =rt, where d is the distance from Sam's car to her home (in miles), r is the speed at which she's traveling (measured in miles per hour), and t is the number of hours she's been on the road. 
Avoid words like "position" (height above ground? sitting down? political situation?) and "time" (5 o'clock? January? 3 minutes since the experiment started?). Never mind that your instructor uses these words freely; you can too when you get a Ph.D.
(d) Variables in text are italicized to tell them apart from regular letters.
7. Explain how each formula is derived, or where it can be found.
Don't pull formulas out of a hat, and don't use variables which you don't define. Either derive the formula yourself in the paper, or explain exactly where you found it, so other people can find it, too.
Put important or long formulas on a line of their own, and then center them; it makes them much easier to read:
The total number of infected cells in a honeycomb with n layers is Therefore, there are 100(101)/2=5,050 infected cells in a honeycomb with 100 layers. 

The total number of infected cells in a honeycomb with n layers is 1+2+ . . . +n =n (n +1)/2. Therefore, there are 100(101)/2=5,050 infected cells in a honeycomb with 100 layers. 
Sufficiently advanced copies of Microsoft Word have an equation editor: pull down the Insert menu, select Object, and then Equation. If you don't have an equation editor, you may either try formulas with tabs and fancy fooling, you may use Excel for matrices if you know how, or you may wish to write the mathematics in by hand. All of these are fine options.
A Caution: Mathematical formulas are hard to do on a standard word processor. If you've got something that's really complicated, feel free to leave space and then write it out by hand laterbut then don't forget to do so! (Like I often do).
8. Give acknowledgment where it is due.
Plagiarism is almost certainly the greatest sin in academiasome fiction writers make plagiarism a motive for murder. It's extremely important to acknowledge where your inspiration, your proofreading, and your support came from. (For example, I thank Mark Stanton, a high school mathematics teacher in New York City, for catching a spelling mistake in the previous sentence.) In particular, you should cite: any book you look at, any compuational or graphical software which helped you understand or solve the problem, any student you talk to (whether in this class or not), any professor you talk to (including and especially me, because I'll catch you if you leave me out). The more specific you are, the better.
While you're at the Writing Center, get a copy of Using Outside Sources and also of (S)he: A Guide to Nonsexist Writing. If you don't need them for this class, you'll need them for another one.
9. In this paper, are the spelling, grammar, and punctuation correct?
(a) It may surprise you that it is on spelling and grammar that people tend to lose most of their points on their mathematics papers. Please spellcheck and proofread your work for grammar mistakes. Better yet, ask a friend to read your paper. Mathematicians are generally not petty, but neither are we amused by sloppy or careless writing.
(b) Mathematical formulas are like clauses or sentences: they need proper punctuation, too. Put periods at the end of a computation if the computation ends the sentence; use commas if it doesn't. An example follows.
If Dr. Crannell's caffeine level varies proportionally with time, we see that where C is her caffeine level t minutes after 7:35 a.m., and k is a constant of proportionality. We can solve to show that k =202, and therefore her caffeine level by 11:02 (t =207) is In other words, she's mightily buzzed. 
(c) Do not confuse mathematical symbols for English words (= and # are especially common examples of this). The symbol "=" is used only in mathematical formulasnot in sentences:
We let V stand for the volume of a single mug and n represent the number of mugs. Then the formula for the total amount of root beer we can pour, R, is R =nV. 

We let V = volume of a single mug and n = the # of mugs. Then the formula for the total amount of root beer R=nV. 

We let V stand for the volume of the mug and n represent the number of mugs. Then the formula for the total amount of root beer we can pour, R, is R is nV. 
(d) Do, however, use equal signs when you state formulas or equations, because mathematical sentences need subjects and verbs, too.
Then the formula for the total amount of root beer we can pour is R=nV. 

Then the formula for the total amount of root beer we can pour is nV. 
10. In this paper, is the mathematics correct?
This is selfexplanatory.
11. In this paper, did the writer solve the question that was originally asked?
So is this.
By the way, these dashes are called "dash" (  ), "endash" ( ), and "emdash" ( ), and they go between words (worksheet), page numbers (pp. 3743), and phrases respectively. So now even your history papers will look better.
In key caps, if you look carefully enough, you can play cards (©®ß™), talk about name brands (Kleenex™, Toys R Us®, Discovering Calculus©), and ask questions in Spanish: ¿Qué pasa?
Whether or not you use a computer, please proofread the final draft before you give it to me. Please, please. No, really.
You are always more than welcome to come see me during my office hours, or to make an appointment to see me outside of office hours. In addition, there is a collection of "excellent papers" from previous calculus classes on the web, so you can see the kinds of papers that I've enjoyed reading.
Feel free to use the Writing Center, even though this is a math class. You can call them at x3866 or drop by the 3rd floor of Keiper (room 316). They have copies of your writing assignments as well as of excellent papers, and they know how to help you organize a paper, as well as what to look for in a good essay. They want to help you. I want them to help you. Get help.
For more information, see Crannell, Annalisa, "How to Grade 300 Mathematical Essays and Survive to tell the Tale," PRIMUS 4, 3 (1994), 193201.