I’ve never had a ponytail palm that did that, but then I never had a ponytail palm for that long. Curious as to where you are located and how you thought this plant was called an “elephant plant” (never heard that one – maybe a geographical thing?)
Anyway, I’d use my trowel to shear the suckers off, trying to catch a little bit of root system and transplant. (Won’t hurt the parent plant as long as you don’t slice the major taproot off.) Once the outdoor temperature is consistently over 65 degrees where you live, that plant can go outside in the sunniest spot you can find. (And give her a little mist with the garden hose 'bout once a week – palms love humidity. That advice stands even if you keep it inside as a houseplant.) Bring back in once temps start dropping below 40 consistently, at night. If you can’t manage to get any roots to come off the parent plant with the pup(s), pick up some rooting hormone at your local nursery or home improvement store. You just dip the bottom half inch of the plant in the rooting hormone (a white powder) and stick it in a pot. Water regularly, a little more at first than you normally would, but don’t let it stay soggy. Palms like to dry out in between waterings. And, hey! E-mail me a picture of it if you can and have a chance… my e-mail is in my profile.
Your resident Florida Gardening Expert.*
*title is self decreed. YMMV