From a discussion on teaching post-calculus mathematics

How do you study for tests with proofs?

On Thu, 26 Jun 1997, Douglas Shaw wrote:

>So I ask this list:  What do is the best way for a graduate student, who
>is making the transition to abstract, proof-oriented, "real" mathematics
>couses, to study for mid-term and final exams of the "prove the following"
The following advice is based largely on what I failed to do right in graduate school.

I think you should force yourself to do the following things:

  1. Pick theorems apart: try to understand why each hypothesis is there. Where is it used in the proof? Can it be weakened or eliminated? Which hypotheses are the more interesting ones? What is a one line summary of the theorem? One of my danger signals indicating that I am not following a text is that I come across a statement which I am willing to believe is true, but realize that if the negation of the statement had appeared instead, I would have been equally willing to believe it.
  2. Work out examples carefully. Try the ones in the book, and try to vary them or make up your own. This is very hard to do. It's much easier to just read the abstract statements of theorems, and forget examples completely. Start by trying out the trivial case -- even that can yield insight.
  3. Practice a lot. In my experience, graduate exam problems are not -that- different in spirit from what you find in a typical problem set in a textbook. The more you try, the better luck you have in an exam.
I say "force yourself" because these are not the path of least resistance through a book or lecture course. The easiest thing is to cruise through books and lecture notes "verifying" that the proofs are valid. It takes an effort of will to criticise the text, work out examples and so on. Thus, although this advice sounds kind of bland and obvious, the nonobvious part is how hard it is to follow.

The more examples you have in your head, the easier it is to understand new theorems and abstractions (and solve problems), because you can "try them out" on your menagerie of examples. I think this can't be overemphasized, and it took me a long time to realize it as a student. I still don't do it enough.

Oh, one more thing:

Yair Minsky
SUNY at Stony Brook

Reading Mathematical Definitions

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