When you read a solution to a problem, you need to
be both respectful and skeptical. Keep in mind that the author
worked hard on the solution and she believes that it is a correct and
well written proof. Most problems in mathematics have more than
one correct solution, so remember that a solution which is
different from the one you found may still be correct. Since most
problems in mathematics have even more incorrect solutions than correct
ones, you also need to be very skeptical when you read solutions.
Ask yourself if you really know that each step of the proof is correct,
that it follows from the previous ones. Remember that even very
famous mathematicians (Fermat, Minkowski,...) have published incorrect
proofs and theorems that just plain aren't true.
Checking for correctness
Check that the author solved the right problem and
that
she didn't transform it into a different problem that is not equivalent
to the original. Make sure that theorems, definitions, and
algorithms are applied correctly. Watch carefully for assumptions
that aren't part of the original problem. Check the logic.
Many students will assume the conclusion of a theorem in their proofs,
they will use an implication backwards, and so on.
Checking that the solution is well written
The problem should be clearly stated, and the order of the
solution should be both logical and clear. The individual steps
should be roughly the same size and appropriate to the audience.
The format, handwriting (or fonts), grammar, spelling, sentence
structure should enhance the readability of the solution, not obscure
it.
Sometimes when you read a proof,
you are left with a feeling that the proof is really cool and
that
you now truly understand the problem. Furthermore, you think that
there couldn't possibly be a better way to proof the theorem.
This
is one of the signs of a good proof. On the other hand, a proof
which is long, hard, and somewhat circuitous will often
leave you with the feeling that there must be a better way to do the
problem. This may be the sign of a not-so-good proof. (It
may also signify that you don't have a good understanding of the
underlying mathematics, or that this is just a hard subject...)
Determining when a proof is good is one of the hardest parts of
reviewing someone's writing.
Making comments
Your comments should be brief, helpful and to the
point. Your job is to point out places where the solution is not
clear or not correct; the author's job is figure out what went wrong
and how to fix it.
Keeping track
Keep track of whose problems (and which ones) you reviewed, which
problems you did, who reviewed them, and whether their comments were
helpful or not. At the end of the term, I will ask you how many
problems you gave to others to review, how many problems you reviewed,
and to grade your reviewers on the quality of their reviews.