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Re: Proofs are so annoying, any shortcuts?
Posted:
Jan 19, 1998 1:02 PM
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A proof is an argument that convinces the reader that what the author is trying
to prove is true. There are no shortcuts because you do not know what the
reader already knows. As you prove theorems -- they become shortcuts. For
example, if you prove right triangles are congruent if they have a hypotenuse
and a leg congruent by using the Pythagorean theorem to show that the third leg
is also congruent and then using SSS, you can now name that as the HL
(hypotenuse-leg) theorem and use it as a shortcut. Thus, most of the theorems
you prove in a beginning geometry course are statements that you would like to
use as reasons in subsequent proofs. If what you are trying to prove is not
labeled as a theorem, but rather as an exercise, then it is probably not as
important for later use or is merely an example of where the theorem you just
learned is useful. Corollaries are mini-theorems that are direct results of
the theorem you just proved.
Now the main question is what consists of a valid argument. Most high school
students learn the two-column method of proof where the steps are numbered in
the left column and the reasons correspondingly numbered and aligned in the
right column. The purpose of this is to organize your thinking. Each step
must have a reason and must logically follow from what is given or a previous
step in the proof. This is sort of like explaining something to a young child
that asks why after every statement. Some people find it easier to think
backward -- that is start with what you want to prove and determine why that
might be true (for example congruent triangles using SSS), and then determine
why the reason for that would be true (for example the three sides congruent),
and so on. And then they write the proof forwards.
No one is born with a talent for writing proofs. It is an acquired skill that
one gets by reading and writing lots of proofs. Unfortunately, many students
don't like to write proofs and so many teachers don't like to teach them. I am
afraid that over the years, we teach less and less proof writing in the high
schools and colleges.
My main suggestion -- read your textbook carefully and then see if you can
replicate the proofs given. Try to prove as many exercises as you can -- even
if they are not assigned. The better you are at proofs, the easier future math
courses, logic courses, etc. will be for you. Unfortunately, there are no
shortcuts to being good at proofs.
Eileen Schoaff
ex-high school math teacher
Buffalo State College
Buffalo, NY
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