You are taking an advanced mathematics class. One of your burdens in a class at this level is to write homework solutions clearly and completely. In this vein, it is important to make a distinction between scratchwork and a properly-written homework solution. Scratchwork is something you produce when you are working through a problem to find a solution. You have taken many classes in which you have learned how to scratch through mathematics problems. In this class, you still have to produce much scratchwork if you hope to work through all of the problems. However, a page filled with nothing but scratchwork does not tell a reader much, and so scratchwork is not worth credit.

Since this is an advanced class in mathematics, it is inevitable that you must therefore use the written language to explain your work. The fundamental rule is that you must write prose to explain your solutions. Every solution should be written in such a way that anyone taking this class who is reasonably prepared can follow and understand your solution. You have studied many things about writing in your English composition classes, and you must apply all of those principles in this class (and elsewhere). You can find examples of properly-written solutions in your textbook. Every "example" in your textbook provides an example, and you are encouraged to mimic the style used in these examples.

There are many things to say about writing homework solutions, but it is possible to give a few highlights. Thus, this document provides a brief summary checklist to help guide writing your solutions. It represents a list of the most important things to consider when writing your solutions. If you do not follow all of these guidelines, then a reader may have significant difficulty understanding your work (and therefore you may not receive full credit). Since this is intended only as a summary, there are some gaps. If you have questions about any of these guidelines or any other ways to improve your writing, therefore, don't hesitate to ask.

• Every solution must follow a natural, organic narrative. Introduce by stating your assumptions, explain the steps required to work the problem, and then make your conclusions.
  
• After you work out a problem and before you write it up, re-read the problem and make certain you have addressed every issue raised in the statement of the problem.
  
• Avoid the use of "narrative arrows". Instead of using an arrow, write a sentence which directs the reader how to follow your solution. (Ask yourself what an arrow is supposed to say, and then write it.)
  
• Avoid the use of "circling your answer". If you are tempted to put a circle or a box around something, then write a sentence which indicates that you are making some conclusion. (Ask yourself what a "circle around your answer" is supposed to say, and then write it.)
  
• Spell words correctly. If you are unsure how to spell a word, look it up in your textbook or a dictionary. Never abbreviate words.
  
• Construct sentences properly. Usually, reading something aloud will reveal problems with how it "sounds". The most common problems are grammatical. Every sentence should begin with a word, and that word should begin in a capital letter.
  
• Use punctuation properly. Every sentence, including every equation, should end with some punctuation mark.
  
• For every equation you write, you should explain its context by justifying it or explaining its relevance.
  
• Use separate paragraphs if you need to organize a lot of ideas. Sometimes each individual step of a problem requires its own paragraph. A line break or an indentation is a common way to indicate the beginning of a new paragraph.
  
• Properly use formal, technical language. The use of the word "plug" is perhaps the worst offense of this rule. The formal word is "substitute".
  
• If you write your solutions by hand, then you must write legibly. Cursive writing is not allowed. (You are encouraged to typeset your solutions!)
  
• Avoid the use of hyperbole. The words "easily", "clearly", and "obviously" are examples of this. Usually the content of your exposition is essentially unchanged if you simply omit these words. (Is anything genuinely "obvious"?)
  
• Don't write sentences in the margins. Every sentence should play a supporting role as part of a paragraph.
  
• If an equation is important enough, then consider writing it centered on its own line. It may even be necessary to number your equation.
  
• If you use an equation or a theorem from the book, then you must specify how to find the result. It's often better instead to use the common name of the result, like "chain rule", "fundamental theorem of calculus", or "existence and uniqueness theorem", etc.
  
• If you number your pages, then it is not necessary to tell your reader to turn the page.
  
• Don't be vague. For example, if after reading a sentence the first reaction is to ask for more information, then you probably need to be more specific.
  
• Use mathematical notation correctly. For some reason, for example, the equals symbol "=" is often horribly misused. (Remember that "equals" and "is" are synonyms.)
  
• It is not necessary or even desirable to spell everything as English words. Instead, use mathematical notation where it is useful, convenient, or required (but sparingly).
  
• If you choose to make up notation (not advised), then you must explain how it works.
  
• Don't submit crossed-out work. In particular, cancellation marks should not appear in your solutions. (Nothing wrong with doing it in your scratchwork, though -- I do it all the time!)
  
• Do not submit scratchwork. Carefully select the equations which are absolutely essential in your exposition and only write those.