mediate
Definition: (verb) to act to reconcile differences between two parties.
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Problem
What is the slope of a line that passes through the points (5, -2) and (-1, 3)?
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Part One
It's a chronic failing of professors that they talk to people as though everybody shared their level of expertise. This can be quite intimidating, or even embarassing; it can make you spend a lot of time explaining that, no, you've never actually read that book, and who is that author again? Nevertheless, most professors don't really expect prospective graduate students to be experts.
Whether you're writing a statement of purpose or having conversations with faculty, it's a good idea to have done your homework. Know the professors in the department you're applying to, and know the major players in the field that you're pursuing. But don't feel that you need to sound like a professor yourself. Stretching your knowledge to make yourself sound wiser than you are is a sure-fire to sound silly. You're better off talking about topics closer to home in which you really are knowledgeable. Remember as well that, often, the best way to display your intelligence is to ask questions.
Part Two
"Never apologize, never explain," runs the classic advice. The idea is that you should never bring up your faults yourself; it emphasizes your flaws, even when your purpose is to justify them or explain them away. This advice goes double for grad-school application. It's one thing to discuss briefly some difficulty you have had to overcome to arrive where you are today; that often creates a positive impression. It's another thing altogether to bring up something you've done wrong, such as bombing an exam or a whole semester, and try to explain it. Never do that.
Stick to the positive when you talk about yourself, whether in person or in writing. You should be prepared to discuss your mistakes if other people bring them up, but don't volunteer them yourself. If you stick to what's good about yourself, the bad stuff may never come up at all.
As far as shortfalls go, such as the fact that you've never studied an important area of your field, think of them as "not yet" categories rather shortcomings. If asked, present them as areas that you're looking forward to doing work in and learning more about; then try to shift the discussion back an area in which you are already knowledgeable. After all, no one is expected to enter grad school already knowing everything there is to know about a field.
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