when you have your dancers pirouette from fourth, do you have them in an open fourth (lined up with first) or more of a crossed fourth (lined up with third? And, in that case, when do you use an open fourth versus a crossed fourth?

We do a crossed 4th ( from third) always.

Generally if they're ready for a pirouette from 4th I teach it crossed, because I generally teach it from a tombe pas de bourree or facing en face and tendu a la seconde and demi rond de jambe to 4th. However when I first introduce 4th in plies I teach it open and cross it once they have control over their weight placement and hip alignment.

I will use a closed fourth. I have a tendency to do pirouettes more from fifth than fourth, because it is harder to turn from fifth and cheat and easier to cheat in fourth (as you know) but once they do, opposite fifth, or third if needed.

I too stress from 5th, but in ATF combos, I tend to do crossed 4th so that the leg muscles (adductors mainly) are engaged. I teach open 4th to my kids ranging in grades 4-6 primarily...

If you try to do both styles of turns yourself you'll feel that you get a much better push off if you use a crossed 4th, open 4th isn't the way.

That is interesting. I don't always specify which fourth I am asking for, even though I do specify that there are two ways to do it. I probably should monitor that better. If I do specify, like during a pirouette, I usually will say open it up more because I think they can feel their weight better and have a more stable base to push from. However, I do realize that is the easy way to do it and that narrowing the base makes the turn more difficult. Like to do a pirouette from fifth, it is much harder, but if they can do it the hard way, they should have no problem with the easier ways. Hmmm...very interesting.

Depends on where they come from to get in fourth.
From fifth to fourth, or with a rond de jambe to fourth = closed fourth.
From first to fourth = open fourth.

I'm with JLL in preferring to have them turn from 5th cuz it's harder to cheat! However, we always use 4th croise--across from 5th, not 3rd--with the weight shifted onto the front foot, because then they just have to releve straight up. In 4th ouverte, they have to shift sideways as well = more potential to lose their center.