This exercise is a classic Pilates exercise. The focus is on using the core muscles to slowly articulate the vertebrae. The back should look round like the letter C throughout the entire exercise. The more you tuck the tailbone under and draw in the lower abs, the more work you will get out of the exercise.
Start sitting up tall on a mat. Feel the crown of the head reach up toward the ceiling and the spine lengthen. Take an exhale as you tuck the tailbone under and slowly roll your back down to the mat one vertebrae at a time. If you are properly rounding your back, the lower back will be the first thing to touch the mat. Inhale as you lay the head and shoulders all the way back, arms to the ceiling, keeping the abdominals engaged. As you exhale, drop the chin toward your chest, draw the abdominals in, and slowly roll back up to a seated position, one vertebrae at a time. (Remember to maintain that “C” curl on the way up.) The legs should stay down, firmly pressing into the mat. It should look smooth and precise. Try not to jerk your body forward in order to make it back up. If it does, you either have weak abs or tight hip-flexors. In that case, you could try the same exercise with bent knees and see if it makes it a little easier.