What is the best way to keep your clothes from shrinking in the dryer:
dry them at a high temperature for a short period of time or dry them
at a lower temperature for a longer period?

Some clothes are much more sensitive to shrinking than others so
perhaps you could clarify as to what you are drying.  I assume when
you say "high emperature for a short period of time" you are taking
the clothes out of the washer while still damp??  If you fully dry
them at a high temp you will get maximum shrinkage.  One thing to
consider is that as long as the clothes are wet/damp there is water
evaporating and thus keeping the temperature lower.  Thus, removing
clothes while damp is a shrinkage preventor whether the temp is high
or low.  Another variable is how full you stuff the dryer.  A stuffed
full dryer will take a longer time to dry and the outside pieces get
real hot while the inside pieces are still damp.

Dry them HALF-WAY at a lower temperature then let them hang dry the rest of the way.

I guess what I'm looking for is if cotton clothing will shrink more if
I dry them faster. Currently, I set my dryer to medium heat and keep
my clothes in there for about 30 minutes. This was the way my mother
taught me to do it since she said the low heat prevented shrinking. I
could cut that time to about 10 minutes on high heat, but I'm curious
is the higher heat (albeit for a shorter duration) would make the
clothes more likely to shrink or if that is just a false assumption.

Too bad we can't sign up to answer.

I've noticed this myself.  I had a problem with all my clothes
shrinking.  At first I thought it was something I was doing with the
washer machine, but after some experimenting I found that putting the
dryer on a cooler setting wouldn't shrink my clothes.

For a more offical response, Laundermac, a company that has designed
washers and dryers since 1940 has this to say
(http://www.askit.com/ai/askit.taf?topic=23):
Shrinking usually occurs when you wash with "Hot Water" and
particularly when you dry clothes in a dryer under the "Hottest"
temperature setting. Line drying will help prevent shrinking!

Dry cotton on high heat, BUT if you're not going to take them out of
the dryer immediately (before you fry them), then put it on the
permanent press setting so it cools down before it stops. Wrinkles get
in your clothes because the clothes are still hot/warm when they've
stopped tumbling.

Your cotton clothes will not shrink more and more, nor do I believe
they'll shrink more due to a higher temperature. They'll shrink to X
extent and that's it.

Follow tutzdad's advice on anything heavy, like woven cotton sweaters
or jeans that you don't WANT to shrink too much (or are really too
tight on you).