Whilst revisiting some old Pilates texts recently, I was reminded that "Joe Pilates didn't like to see people make abrupt motions with their backs, especially with their backs straight." This is where the 'roll your spine vertebra by vertebra' cues come about, to roll and unroll.  

How many of us when we do a roll up (from lying flat on the floor to a seated position), throw our arms up and down or thrust ourselves forward with momentum?  Sure, that's one way of getting there but can you be gentler with your body?   Think about how a wheel rolls - it's smooth (depending on the ground surface).  Can we do a roll up like that?  Smoothly until we cannot go further, and then use our arms by pushing off the floor to continue upright.  Rolling back and forth is one way of rolling but there's also side to side and even varied angles to roll on.  

Explore ways to roll to help the spine become more flexible and move smoothly.

Reference:  Return to life through Contrology" by Joseph H Pilates https://babel.hathitrust.org/cgi/pt?id=mdp.39015038069475&view=1up&seq=41