Guidelines for Written Work

1.The Purpose of Written Work

The purpose of writing mathematics is to develop skills in understanding and communicating mathematics. In a class you communicate with your teacher, with classmates and with yourself (when you go over your work later). If you use mathematics on the job, you will be communicating with your fellow workers, your boss and with yourself. It is important that your communication be clear so that another person can follow the steps you have taken. We are all familiar with the fact that we are less likely to make mathematical errors when we write things down than when we work "in our head." The more carefully and clearly you write your mathematics, the more likely it is to be correct. Finally, writing mathematics carefully helps you to understand the mathematics better. There is a lot of truth in the saying "If you can't explain it, you don't understand it."

Do not think of your homework paper as a certificate proving that you have done the assignment. Think of it as an exercise in learning and in reporting what you have learned.

2. Your Responsibility

Communicate with the reader. Do not write to the instructor (who already knows how to do the problem), but explain your solution to someone who needs help, perhaps a classmate who has been sick. Start at the beginning and be clear, logical, and complete.

The ultimate test of what you write is this: can someone learn from your paper easily ? Remember, the reader will see only what you wrote, not what you meant to say, so it must all be there, and be accurate.

Make your paper "reader-friendly."

3. Common Errors in Written Mathematics

Below are some common errors students make when writing mathematics:

incorrect math	not defining terms
bad English	using the same symbol for different things
abusing the equal sign	not answering the question
using bad format	omitting reasons

To help you avoid these errors, let's discuss each one and give some examples.

A. Use correct mathematics

Mathematics is the most precise of all subjects. Every statement you make must be correct. A solution will be ruined by a single false step such as

.

If you are not sure about a statement, check it for some special cases. For example, if you suspect that n² > 2ⁿ, try a few values of n. You will soon see that 25 < 2⁵, so the statement is false.

Be very careful not to make false assertions.

B. Use your best English

In mathematics we frequently use mathematical symbols instead of English words, but a string of formulas with no explanations in English can be impossible to understand. Good mathematical writing contains the right balance of words and symbols. Even when symbols are used, the rules of grammer apply as indicated below.

Good communication requires good English. A correct solution garbled by bad English may be worthless to the reader. The rules of grammar, spelling, and punctuation apply to mathematics.

For clear communication, present one idea at a time. Since an idea is expressed by a complete sentence, write in complete sentences.

Examples:

Nonsentence	Sentence
a + b	a + b = 15
since x is positive	Since x is positive, x > -2.
(x+y)^2	(x+y)^2 = x^2 + 2xy + y^2 .

Use words precisely; avoid ambiguity. Vagueness and ambiguity are incompatible with mathematics. Learn the correct technical words and learn to use them.

Bad	Good
This equation never seems to cross the x-axis	The graph of the function never seems to cross the x-axis...

Keep your sentences short. Long sentences are hard to follow and tend to become garbled. Break up long sentences into shorter ones.

Avoid ambiguous pronouns (it, this, that, which). Pronouns are often not precise and they often cause confusion. Always consider whether substituting the actual thing for the pronoun will make your statement clearer or more forceful. Pronouns are okay only when its perfectly clear what they stand for.

Bad	Good
To maximize it, differentiate it then set it equal to 0, which when you solve it, it is the maximum.	To maximize f(x), solve f'(x) =0. Let xo be the solution. Then the maximum is f(xo).

To avoid ambigous pronouns, give names to quantities and use those names. Don't maximize "it", maximize f(x).

Do not use arrows. They are a lazy person's way of avoiding writing well, and they confuse the reader.

Use words and symbols as appropriate. It is difficult to read mathematics if only words or only symbols are used. Symbols are a great shorthand for mathematical statements or calculations, but words are needed to introduce and explain the direction and purpose of the computations.

Bad	Good
Let x=2^n + 2^n, a power of two plus itself, doubling it or raising two to the next power.	Let x=2^n, then x+x=2^n + 2^n=2(2^n)=2^n+1.

C. Honor the equal sign.

Quantities on either side of an equal sign must be equal.

Oranges do not equal apples, numbers do not equal sets. The equal sign has a precise meaning; it is not a punctuation mark such as a dash--.

Bad	Good
n = even = 2n	If n is even, then n = 2k for some k.

[Note: n does not equal 2n.]

n2 = l6 = �� 4

n2 = l6; hence n = �� 4.

[16 does not equal �� 4.]

An n-gon = (n-2)180^o

The sum of the interior angles of an n-gon is (n-2)180^o.

[A polygon does not equal a number of degrees.]

Case l = odd

Case l: n odd

cos A =0.5 = 60^o

cos A = 0.5 , so A = 60^o

D. Use different letters for different things.

For example, write an equation showing that n is a power of 2.

Wrong	Right
n = 2^n	n = 2^k for some k.

In fact, there is no whole number n for which n = 2ⁿ :

An odd number can be written as 2k+l. How do you write two odd numbers n₁ and n₂?

Wrong	Right
n1 = 2k + l, n2 = 2k + l	n1 = 2k1+ l, n2 = 2k2+ l

The k's will be different unless n₁ = n₂.

E. Identify Variables, i.e. define your terms

Whenever you introduce a variable (or a constant), carefully identify what it stands for.

Problem: If the perimeter of a rectangle is 18 cm and the area is 20 cm2, what are the dimensions of the rectangle?

Bad	Good
2x + 2y = 18	Let x = length of the rectangle and
xy = 20	let y = width of the rectangle.
so x(9-x) = 20	Then 2x + 2y = l8cm,
x^2 - 9x + 20 = 0	xy = 20cm^2
(x-4)(x-5) = 0	etc.
x=4 or x=5	etc.
dimensions are 4 cm and 5 cm	Hence, the dimensions are 4cm and 5cm.

A labeled diagram often helps in identifying variables.

Expressions such as "let t be the time (in seconds)," "where n is an integer," and "for any real number x" are often used to identify variables.

F. Give reasons.

Problem 2: Prove the Pythagorean theorem.

Proof.

Let a right triangle have legs of length a and b, and
hypotenuse of length c. Then a² + b² = c². (End of "proof.")

Comment: The writer gives no reason for the conclusion. He just
states the theorem without proving it.

Problem 3: Solve the equation x³ - 15x² - 28x - x = 85.

Solution: First combine the two terms in x:
x³ - 15x² - 29x = 85.

Now subtract 85 from both sides.

x³ - l5x² - 29x - 85 = 0

Hence, x=l7.

Comment: Wow! Where did x=l7 come from? Out of a hat? Is the writer a genius? This student explains two very easy steps that hardly need explanation. But when she comes to the heart of the problem, she pole vaults over it.

Use common sense. Indicate the main points in a problem. Help the reader through the hard parts.

G. Be explicit.

Problem 4: Can an even number be divisible by an odd number?

Vague	Explicit
Yes, there is nothing to prevent it.	Yes, for example, 10 is divisible by 5.

Problem 5: Are there any fractions of the form 2/n that are less than 0.01?

Vague	Explicit
Yes. You can show that if n is large enough, then .	Yes. Take n=2000. Then

Comment: Don't say that something "can be done." Do it.

H. Answer the question.

When you finish a problem, go back and read it again. Be sure you have given a clear answer to the question asked.

Problem 6: I drive from here to Chicago, 125 miles at 50 mph, then go from Chicago to Minneapolis, 550 miles at 55 mph. How long does it take me to drive from here to Minneapolis?

Nonanswer	Answer
125 + 550 = 675	The time to travel from here to Chicago is 125 miles/50 mph = 2.5 hours. The time from Chicago to Minneapolis is 550 miles/55 mph = 10 hours. Thus, the total time is 12.5 hours.

Problem 7: Explain why the sum of two consecutive integers is not divisible by 4.

Nonanswer	Answer
l + 2 = 3, not divisible by 4 2 + 3 = 5, not divisible by 4	Let n and n+l be two consecutive integers. Their sum is n + (n+l) = 2n + l which is odd, hence not divisible by 4.

[Two special cases do not prove a general statement]

I. Be aware of format.

For example, don't cram. It's silly to squeeze your whole assignment into a couple of inches at the top of the paper, and then leave the rest of the page blank. Look for excuses to leave some space. Break up long paragraphs. Write equations on separate lines. Be considerate to the reader. Make your paper pleasant to read.

Problem 8: Find the maximum area of a rectangle whose perimeter is 100.

Solution 1. A=x(50-x) = 50x-x² dA/dx = 50-2x, 50-2x=0

x=25 Hence A=25(50-25)=625.

Comment. The mathematics is correct, but the presentation is terrible. The solution is so crammed it's hard to read. Terms are undefined and no explanations are given.

Here is a better solution:

Solution 2. The area of the rectangle is A=xy, where x is the length in feet and y is the width in feet.

Since the perimeter is 100 ft.:

2x + 2y = 100

so x + y = 50

and y = 50-x.

Hence, A = x(50-x) = 50x - x².

This function is differentiable for all x, so it can have a maximum only where its derivative is zero. Set dA/dx = 0:

50 - 2x = 0 gives

x = 25.

Because d²A/dx² = -2 at x=25, this value of x yields a maximum for A.

The maximum of A is:

A = 25(50-25) = 625.

Thus the maximum area is 625 ft².

(Compare the use of units in this correct style with that in the example for part E.)

J. Be clear about what is given and what is required.
This is especially important for proofs, but it applies to all problems.

K. Use notation carefully.

It may seem a bother to worry about the details of notation, but it pays off in the long run. Correct use of notation can make problems much easier and incorrect notation can cause muddleheadedness.

Bad	Good
f(x) = x^2 f'(x^2) = 2x (actually f'(x^2) = 2x^2)	Since f(x) = x^2, f'(x) = 2x.

4. Good Writing

Students often say "I understand the math, but I can't write it." Like most people, they don't realize that good writing is not easy and requires practice.

Keep this in mind: Almost nobody can write anything well the first time. Even professional writers constantly revise and rewrite. Therefore, do not submit your first draft. First, work on scratch paper (use lots of scratch paper). Then rewrite using these Guidelines. Here are three ways to check your writing:

1. Put your paper away for a few hours or until the next day. Then read it pretending you've never seen it before. Does it make sense? Can you follow it easily?

2. Read your paper out loud. This is a great way to catch errors and garbled writing.

3. Have another student read your paper.

In short, read over what you write.
The principles of written work expressed in these Guidelines will probably require more thought and care than you have put into your assignments in previous mathematics courses. But they will make you a clearer thinker and a better student.

REFERENCES

1. L. Gillman. Writing Mathematics Well. Mathematical Association of America, Washington, D.C., 1988.
2. W. Strunk and E. B. White. The Elements of Style, third edition. MacMillan, New York, 1979.

These Guidelines are an adaptation of the article: "Learning Mathematics Through Writing, Some Guidelines" by J. J. Price, The College Mathematics Journal 20 (1989) 5.