The HIT school tends to say that the key is to get stronger and then learn to apply that strength explosively if that is what you sport requires. Luke Carlson in my interview with him stated it like this:
A last question – what about explosive training? This is another tactic that often comes up – “train explosively to improve speed and power”. So we see people doing Olympic lifts, bounding, doing depth jumps. Is this just a waste of energy or is there any real benefit?
I think every athlete should train explosively; but this does not mean they should perform Olympic lifts, depth jumps, etc. Improvements in "explosiveness" are stimulated by the INTENT to move explosively; the outward demonstration of fast movement is not important (or desirable). For example, if a trainee is performing the barbell bench press, he/she should perform the initial repetitions in a slow and controlled manner. This minimizes momentum and maximizes muscle tension; muscle tension is the most important element in muscle fiber recruitment. As the lifter begins to fatigue, he/she can in fact attempt to lift the weight as fast as possible - attempt to "explode" through the weight. However, the weight will not move fast because the trainee is fatigued and the weight is heavy. However, from a motor unit/muscle fiber standpoint, the "explosive" stimulus has been provided. This approach can and should be applied to all exercises/muscle groups. If the weight actually moves fast during strength training, momentum is introduced and muscle tension is reduced (as the musculature is essentially unloaded); this is the exact opposite of the goal of strength training and the requirement for muscle fiber recruitment.
Anyway, that is just introduction.
Traditional sports science has also promoted explosive moves. However I spotted this abstract yesterday and the full paper is available here.
A Comparison of Ballistic and Non-Ballistic Lower-Body Resistance Exercise and the Methods Used to Identify Their Positive Lifting Phases.
As I read it the researchers are saying that one of the issues with the traditional approach is that the way in which the metrics were measured skewed the results. The usual methods of measuring power made the ballistic exercise seem to produce more power.
No significant differences were found in mean force or mean power between ballistic and non ballistic exercises.
Their conclusion is that
These results challenge common perceptions of Ballistic superiority for power development.