Neurological Component of Core Stability
Often times the neurological component required in core stabilization is neglected in training. In reality, muscles are actually "dumb" because they only respond to commands sent by the brain and the spinal cord. These signals are sent from the brain and spinal cord (central nervous system) via motor units which attach to the muscle itself. Core stability is rooted in the proper timing and activation of core and postural muscles before the actual intended movement occurs. This concept goes back to my first post in that core stability is not a product of pure strength; and importantly, training weakness won't always solve the problem. These timing sequences are stored in the body as movement patterns. Storage of movement patterns in the brain creates efficiency for the body and reduces processing time of the brain by categorizing single movements into groups of movements. This means that isolated abdominal crunches won't do the trick if you want to stabilize the core. They will strengthen the abs (which we all need!), but it isn't the best choice for training your core. Instead, do movements that challenge the muscles the way they're designed and expected to work in real life. Isolation work is good for strength, but not necessarily the answer for a strong core.
The second highlighted quote was just cool because it explains why rolling like a ball is not a throw-away exercise! So to all my clients, that's why we do it!