If your goal in completing homework is to maximize your learning and the grade you receive while minimizing the time and effort you expend, then read on--this page is for you.

• Identifying an Audience
• Writing with Clarity
• Using Symbols Well
• A Simple Example
• A More Complex Example
• Saving Time
• A Checklist for Evaluating Your Homework

Identifying an Audience As with all communication, it is impossible to create effective homework solutions unless you have identified a target audience. Your goal is to convince your professor that you really understand the assigned concepts. You don't want to skip too many details and assume that he or she believes you know the answer, but you also don't want to bore your reader with trivial details. The trick is determining what the trivial details are.

A good way to figure out what to include is to assume that the reader of your paper is a student in your class who is a few days behind so they haven't yet done the homework. It is often helpful to keep some particular person in mind while writing. If you were trying to convince the student who normally sits to your right of the correctness of your answer, what would you need to include? If the student you imagine would need to see the equations you've written down simplified in order to understand your solution, then do the simplification. If you are using terms or symbols that haven't been defined in the course, then define them. If you are in a senior honors course in analysis, then describing the details of a routine computation involving integration by parts is unnecessary detail, but if you are in a first semester calculus course, then it would be unwise to assume your reader is comfortable filling in the details of such an integration. To maximize your grade on homework, your solutions should contain everything necessary to make them easily accessible to the student you imagine. To minimize your effort, don't include details about topics your reader already thoroughly understands.

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Writing with Clarity Mathematics is often difficult to read, but that should be because the concepts being conveyed are difficult, not because insufficient time has been spent developing the exposition. No one can sit down and write a good essay on a literary work without going through several drafts, and the same is true for a paper written about mathematics--it will need to go through several drafts. Often, you will use up many sheets of paper and many hours or even days with false starts and speculations before finally creating a draft of your answer. So that your work might get the recognition that it deserves, it is essential that these initial explorations are followed by carefully crafting a paper that documents the results of your efforts.

Tailor your response so that it is self-contained. The reader of your paper should be able to determine what you want to say without referring to a textbook or assignment sheet to find the statement of the problem. When beginning a proof, state your assumptions, explain your notation, and describe your goal. If the proof uses contraposition, contradiction, or induction, then be sure to include that information up front. This will not only help your reader, but it will help you. Clearly mapping your proof strategy, assumptions, and notation, will highlight errors in your logic and help ensure that your work is understood and appreciated.

To ease comprehension, write your work in complete sentences and use periods and commas as you would with any writing. While complex and varied sentence structure add interest to prose writing, simple declarative sentences are usually better in mathematical writing. Typically, the reader will pause at the end of each sentence to sort out how the idea contained in the sentence follows from the preceding work, and an idea contained in a shorter sentence will be easier to parse than a complex web of implications contained within a single sentence. Since mathematicians are trained to look back for justification, if you make a statement that you intend to prove later, you must indicate that the proof has been deferred, either in the same sentence or a prefatory statement. Without such prompting, your reader is likely to waste time trying to figure out why the statement follows from the earlier work, which will undoubtedly cause frustration. (And a frustrated instructor will be much less likely to look kindly on your work than one who breezed through it easily.) The idea is to break your proof into small steps, each of which follows from the preceding steps and is presented in its own sentence. If your proof involves many steps, then break it into paragraphs each of which represents a larger step. The Division Theorem has a long proof, which is best broken into chunks.

Since it is often necessary for readers to refer back to earlier material, it is usually best to write on the front of the pages only. It is very difficult to compare two equations if one is on the front of a page while the other is on the back. If you are writing the solution to a problem and get to the bottom of the page, don't try to squeeze in the rest of the solution; go on to a fresh sheet of paper. And don't rotate the paper to extend your equations up the side. Nothing is more irritating than reading a sentence that turns the corner and starts up the page margin! If you want to conserve trees, then use the backs of your work for scratch paper. Incidentally, using only one side of the page is good practice when doing the preparatory drafts, as is limiting the amount of information you place on any one page. This facilitates making comparisons between various versions of your work and allows room for adding comments later.

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Using Symbols Well Proper use of symbols will bring a refreshing clarity to your work, but poor choices will befuddle your reader--the trick is to sort the good from the bad. While we won't try to unravel all of the intricacies here, there are few general rules worth observing:

• If you don't know why you are using a symbol instead of a verbal description, don't use one.
• If you aren't going to refer to something more than once or twice, don't introduce a symbol to describe it.
• Never change the name of an object in the middle of an argument. (While it might be logically correct to start calling x y, what would be the point?)
• Define every symbol, and do so before you start using it.
• Always use different symbols for different objects. For example, if you have two different functions, don't call them both f.
• Choose symbols that are consistent with mathematical convention. There are no hard and fast rules, but if you look for it, you will notice a remarkable consistency. For example, real number variables are usually denoted by u, v, x, and y; real number constants are often denoted by a, b, and c; integers and natural number are typically denoted by k, n, m, p, and q; and functions are often named f, g, or h, with p and q often reserved for polynomial functions. The best way to learn the conventions is to pay attention to how symbols are used in class and in your reading.
• After you've finished your writing assignment, make a list of the symbols you've used to ensure that they are necessary, well-defined, consistent, and agree with the usual conventions.

1. Every set S of natural numbers contains a smallest number. (S is unnecessary.)
2. S contains a smallest number. (S isn't defined.)
3. For every positive real number f, there is a positive real number that is less than f. (The symbol f is used in an unconventional manner.)

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A Simple Example To put this all together, suppose we want to prove that when the product of two numbers is odd, then the numbers are odd. This is easily done by contraposition. That is, it is easy to prove the logically equivalent statement that if at least one of two numbers is even, then their product is even. Note that while most readers would understand from the context that the numbers in question have to be integers, we can improve the clarity by specifying that they are integers. Since it is easier to discuss two numbers if they have names, we introduce notation to assist us, calling the numbers a and b. We put this together in a proof as follows:

Saving Time Mathematics is most efficiently learned in small doses, so it is wise to set aside time each day for homework. Often if you spend time on a problem one day, you'll find that when you return to it the next day, or maybe the day after, the solution is clear, whereas any amount of time spent working on the problem in one sitting will not lead to a solution. So three hours spent on a problem over three days will often result in a better result than six hours spent on the same problem the night before the solution is due.

Before you begin writing, look over your assignment and make sure you know what is required. Will a simple answer suffice or is more explanation is needed? Does the instructor require only that you convince him or her that you understand the material, or should your response be sufficiently complete to convince one of your classmates? A few minutes spent identifying goals can save you from writing more than is necessary or a low grade based on an insufficient response. If you have questions about the requirements, ask for clarification. Most instructors would prefer to answer questions up front, rather than having to sift through mounds of writing responding to the wrong problem.

In other words, start early, revisit your work frequently, and focus on your goal.

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