Question about comparatives

[kutsuwamushi.livejournal.com]

*THUD*

Test in two hours. Procrastination - bad. Missing class - bad. Cramming much.

If anyone is out there, can you answer a small question? When using the comparative form of an adjective, does the sentence *need* to contain the thing being compared against (I don't know the technical term; sorry)?

For example:

Моя мама красивее, чем твоя мама!

I'm fairly sure that that's correct. But what if a dialogue is taking place?

-Моя мама очень красивая.
-А я мама красивее!

The first speaker's mom is implied but not stated in the second sentence. Is this possible?

And what if you want to say "I cook better than I used to"? Would you use a comparative adjective, or is a completely different form used?