Males are neutered, females are spayed. Whether or not you do it at all is a very personal decision and there are lots of debates on this board as to which is better for health reasons- but it also comes down to if you're willing to deal with a bitch in heat, taking the proper safeguards, etc. I'd suggest doing some research on those threads and see how you feel.
Whichever way you decide to go, the common thread of thought is that six months is too early to do it. A lot of vets recommend this because many, many (many) pet owners don't know or care to be responsible for an intact female and they sexually mature before they're adults. You really should wait until she is older and done growing.
Delta zur Bindenburg- 11/15/13
Gentle Tripp- 6 yr old OTTB mare