M360M/396C, Fall 04

GUIDELINES FOR WRITING UP PROBLEM SOLUTIONS

Writing is important in this class. First, writing helps you clarify your thinking. Second, a teacher needs to develop the skill of explaining mathematics clearly. In particular, writing clearly can help you learn to write clear assignments, develop good assessments, and explain orally as well as in writing. Here are some guidelines to help you in mathematical writing.

I. Audience

Your audience for your written homework in this class is the entire class plus the teacher. This means that your writing should "speak to" this entire group of people.
Be careful not to assume that your reader knows anything beyond what is on the sheet "What You May Assume ..." (and any additions made to it) and anything previously discussed in this class. In particular, you may not quote results from advanced math classes.
Assume your reader has not worked on the problem.
Assume your reader wants to know why as well as how and what:

-> wants proof, rather than just examples, that something you claim to be true in general is indeed true in general.
-> wants to follow the logic of your reasoning.
-> wants to understand conceptually as well as follow logically (so would appreciate examples, in addition to proof, if they help in understanding).
-> wants to know the main points (but not too much gory detail) of how you figured out your solution: what led you to take the path you did?

II. Process

You can’t expect to have a good write-up on a first draft. Write out a draft, then revise to make it better.

-> Check for correctness of mathematics, including reasoning.
-> Reorganize and reword to improve clarity and flow of explanation.

For some people, first drafts are often wordy. Others go in the opposite direction, writing only equations or pictures with no words to help explain. Try to find a balance of words, equations, and pictures that is best for the particular problem.

-> Be precise, not vague.
-> Put the paper aside, then reread it as if someone else had written it. If you have trouble following, so will your reader, so revise some more.
-> If possible, have someone else read and critique the write-up. (The student who consistently had the best write-ups of any student I have ever had in this class said she always tried them out on a friend who had little math background.)

Writing well takes time, so be sure to allot enough time for it. Plan to have your problem solved at least a day before it is due, so you can spend time writing.
I'm fairly tolerant of having a few typos (I make them too! For example, it took me several times reading this to realize that I had "to" rather than "too" in the last sentence.). I consider the items above and below much more important.

III. Medium

Remember that a written solution and a blackboard presentation of a solution are two very different media. At the board, you are communicating in real time, can draw as you go, and can use both written and oral communication. By all means take advantage of these possibilities when you are at the board, but remember that they are not available for written solutions. In particular, several drawings may be needed in written homework to communicate what could be put on one drawing in stages on the board.

IV. Symbols and Terminology

   Be sure to define any symbols or terminology that you introduce. The reader is not a mind reader, so you need to state clearly what you are letting stand for what. The most common ways to do this are by a sentence beginning with "Let ... ," or a phrase beginning with, "where ..."

   Examples:
        Let x be the length of the side of the square.
       Let v₀ denote the initial velocity.
       Let u, v, and w be as shown in the diagram. (Note: This is acceptable only if there is a diagram with parts clearly labeled as u, v, and w!)
       F = ma, where F is the force, m is the mass, and a is the acceleration.

    Example: State the Pythagorean Theorem.

Incorrect	Correct
a² +b² = c²	a² +b² = c², where c is the length of the hypotenuse of a right triangle, and a and b are the lengths of the legs.

V. Vocabulary

Here are some common misuses of words that are important in mathematics.

One area where some students make errors is in expressing logical consequences. "Therefore", "thus", "hence", "then", and "consequently" assert that what comes next is a logical consequence of what was stated previously. "Let ... " should not be used when something is a logical consequence of something else.

       Example:
           Incorrect                                Correct
           Let x = 2. Let x² = 4.            Let x = 2. Then x² = 4.

"Since" and "because" say that what you are about to say is a reason for something else. The something else might be stated before or after the reason, depending on the sentence structure.

       Examples:
                    Since x = 2, x² = 4.
                    x² = 4, since x = 2.

VI. Grammar, Spelling, and Punctuation
   Try to use good grammar and correct spelling and punctuation. A teacher should set a good example! (Note: I will make appropriate allowances for students who are not native English speakers, but do expect them to pay attention to the points below.)
   I will pay special attention to the following points of spelling, punctuation and grammar that are especially relevant to mathematics:

Be sure mathematical terms are spelled correctly.
Be sure mathematical symbols are used correctly. One symbol that is often misused is "=". Remember that it stands for "equal," or "equals," or "is equal to," or "are equal to." The following are examples of common misuses of the equal sign.

            i. 2x + 1 = 3x -4
                    = 5 = x    (This says that 3x - 4 = 5 -- certainly not what is intended.)

          ii. n = even       (The writer presumably means "n is even".)

   3. Be careful not to use pronouns ambiguously. Watch out especially for "it."

        Example of poor (confusing) usage:

"To find it, multiply it by three, then add two to it."

The word "it" stands for three different things in this sentence! The reader may need to be a mind reader to understand what the writer intends. And writing this unclear often indicates that the writer is confused, too, or will be soon.

Example of good (clear, unambiguous) writing:

"To find f(x), multiply x by three, then add two to the result."

4. Be careful not to use the same symbol to stand for two different things in the same solution.

        Incorrect: Since m and n are both odd numbers, m = 2k + 1 and n = 2k + 1.

        (Does the writer really understand, or have they just mindlessly applied a rule?)

        Correct: Since m and n are both odd numbers, m = 2k + 1 for some integer k and n = 2l + 1 for some integer l.

5. Use logical connectives (since, therefore, thus, because, etc.) correctly.

Incorrect: Since x = 2, therefore x² = 4.
Correct: Since x = 2, x² = 4.

   6. Punctuate logical connectives correctly. It helps to be aware of two types of connecting words:

   a. Subordinate conjunctions begin dependent clauses (clauses that can’t stand alone as a whole sentence). Subordinate conjunctions include:

       since, because, if, when, before, after, until, while, although, unless, that, so that, in order that, as if, though, even though, whereas

A clause starting with one of these words is a sentence fragment if it stands alone.

       Incorrect: x² = 4. Since x = 2.
       Correct: x² = 4, since x = 2.

   b. Conjunctive adverbs introduce new sentences. If you find one in the middle of a sentence without a semicolon before it, you’ve probably got a run-on sentence or comma splice. Conjunctive adverbs include:

therefore, thus, hence, consequently, besides, moreover, furthermore, however, likewise, similarly, accordingly, still, however, nevertheless, otherwise, afterwards, later, earlier, indeed

       Incorrect: x = 2, therefore x² = 4.
       Correct: x = 2. Therefore x² = 4.
       Correct: x = 2; therefore x² = 4.


Please note: These guidelines are intended for written homework only.
   On exams, you will not have the time to revise as much as on homework. I will expect you to give complete explanations (clearly enough explained so that I can follow them without too much difficulty) in order to receive full credit for an exam problem, but won’t pay as much attention to organization, wordiness, grammar, etc. as on homework.
   Similarly, these guidelines are not intended for journal writing. There, the important thing is to express your thoughts clearly enough so that I can follow them. I don’t expect you to write multiple drafts. Of course, if your first draft is illegible, please rewrite it legibly. And if you’re not satisfied that your first draft says what you are trying to say, feel free to throw it out and try again. There is no obligation to give me a first draft.

One more bit of advice that applies to oral as well as written communication: Be very cautious about using words or phrases like "obvious," "obviously," "clearly," "as anyone can see," … These have several potential pitfalls:

Many mistakes occur exactly at the places where someone uses these words.
Places where you use these words may be exactly where you need to be careful to explain to your audience what you are doing.
If you say something is obvious, clear, etc. but it is not obvious to the reader or listener, they may feel put down. This is not a desirable outcome either in this class or when you are teaching!