Hmmm .... this is a very interesting thread.
I must have enough sand mixed in with my soil that dividing mine is pretty painless. I dug up a bunch of them today to transplant to the new raised beds. I used a pitchfork and just loosened around underneath them and lifted them. Shook off as much dirt as I could over the area they were in. Then took the clumps over to a grassy area near the new garden beds and basically just beat them on the ground to remove more dirt. I literally beat them on the ground rotating the root sides until the dirt is pretty much shaken off the roots. I am holding them near the bottom of the leaves (where they are joined to the roots) with both hands and beating them by lifting and lowering the root parts against the ground. I am not gentle. As the dirt falls away from the roots, they start separating. And then I jiggle/wrestle each fan until the roots comes lose from the clump. It only takes me a few moments to do an entire 6-10 fan clump. I always make sure the dirt is dry when I dig them up. If it is wet, the dirt is harder to shake off and to separate each fan from the clump. Works like a charm for me! I've been doing it that way for years.