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From a discussion on teaching post-calculus mathematics

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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"
>type?
>

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:

- 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.
- 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.
- 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:

- Talk a lot with other students. A lot of mathematical thinking
is enhanced by being verbalized.

Yair Minsky

SUNY at Stony Brook

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