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Sound absorbtion behind drumkit?

I have an 8x5 foot American flag behind my drum kit.

I was thinking I can put some kind of sound absorber behind the flag. I figure the flag absorbs an infinitesimal amount of higher frequencies.

What can I "hide" behind it that might take some more ring out of the cymbals while allowing the toms and snare to have some air?

Should I have the Drum kit on a wooden surface for recording

I'm after any advice for my upcoming drum recording.
The room is carpeted and is actually a classroom.. and sounds like a classroom too!
I still believe I can get a good result if I'm careful. I was wondering if it's worth placing the drums on solid wood rather than just carpet.. thinking it might brighten and focus the sound a little.