You can't really train the 'core' unless you know what it is and does. First of all, it is commonly described as the (many) muscles that connect the hips to the shoulders. Sounds simple. However, this is where all of the confusion begins. The Rectus Abdominis connects the Pubic bone to the rib cage. Therefore, it would be considered a 'core' muscle.
But, using this definition, the Latisimus Dorsi should be considered a core muscle, too; it runs from the lumbar spine and hips to the Humerus. But, most people wouldn't consider the Latisimus Dorsi a 'core' muscle.
So, anatomically, there are multiple muscles, all with differing functions, depending on the body's position and the specific movement. What I am getting at, is that what most people think of core training is not very effective and sometimes a waste of time.
You can't and shouldn't view 'core' training in isolation. Just like no person is an island, 'core' muscles work in conjunction with limb muscles in a manner that is specific to a movement.
We need to look at the function of all of these 'core' muscles. Like I mentioned earlier, there are many muscles with numerous roles and activation possibilities. Collectively, what do all of these muscles do? In the big picture, they provide the body with joint stabilization for which limb muscles can act. They do flex, rotate, extend, and bend the spine, but this is a reductionist (very simple) view of how these muscles work. Just because the Rectus Abdominis can flex the spine (see above picture), that doesn't mean that is its primary role. Like all muscles, they work in conjunction with other muscles. The Rectus Abdominis more often functions to resist spinal extension and rotation as the arms and leg muscles contract.
How many activities do you perform without your arms or legs in contact with the ground (such as with a crunch)? Very few. By performing the 'core' exercise, you are reversing the function of the Rectus Abdominis. Now it is the prime mover and not a torso stablizer. It is being used and perhaps loaded (has resistance), but there are a few problems and this is the main dillema with 'core' training.
The first issue (similar to unstable surface training) is limited ability to load these 'core' muscles. Now that the 'core' muscles are being used as a prime mover of the lumbar spine, there is a lack of stabilization of the lumbar spine. In this situation, the Rectus Abdominis' ability to develop force is limited, as is the training effect on this (and other 'core') muscles.
The second, and probably most important issue, is specificity. Since most activities are performed with the arms, legs, or a combination of the two, the 'core' muscles need to all develop tremendous stabilizing forces to counter the powerful pull of the limb muscles. The 'core' muscles work together in a sophisticated integration with the limb muscles. Additionally, some muscles, like the Latissimus Dorsi, both stabilize some joints and move others (for example, during a pull-up).
'Core' training is a popular buzz word in fitness these days. It is promoted as a key component in an exercise training program. However, a misunderstanding of human neuromuscular function or the possible novelty of some of these 'core' training exercises has led many exercisers to waste their time with ineffective exercises. Many of these 'core'-specific exercises do not transfer over to other exercises, sports, or activities of daily living that are performed with the limbs on a stable surface. Exercisers can make better progress by performing resisted exercises that involve the arms and legs with the 'core' muscles "tying" them together.