In addition to the homework hints and exam hints,
Math 199 students should consider the following:
 Outline your proofs before you write them. You should work out the details of your proof before you write up your final version; don't just start writing. This is the same as it would be in any writing that you do: you should know where you are headed and what you plan to say before you start. In class, I may write proofs directly at the board without showing you an outline, but that's because I've already done the thinking that goes into the process. In many ways, however, the thinking that comes before the proof is the most important part  the proof simply codifies this thinking. You can't skip right to the proof without doing the preparation. Remember that you can work both from the beginning (the information that you know) and from the end (the place you are trying to get to).
 You should be able to interpret new definitions on your own. Since our goal to to be able to understand the meanings and implications of mathematical statements, you should be able to figure out the meanings of, and prove things about, definitions that you've never seen before, even without examples in class. So it would not be unreasonable on an exam or problem set to give you the definition of a property that you've never seen before, and ask you to analyze it and prove some simple consequences.
 Translate between notation and words and vice versa. If a definition or theorem is given in terms of words, you should try to rewrite it in terms of the associated mathematical notation. Conversely, if a definition includes mathematical notation, you should translate it into the corresponding concepts in words. (And I don't mean just a symbolbysymbol transliteration, but a conceptbased conversion.) Most students find the translation between the ideas in their head and the mathematical notation on the page to be a difficult one, and this is one that you need to practice carefully.

