I forgot to mention that you can also take cuttings horizontally, but this is not nearly as reliably successful as vertical divisions. As noted above, roots with a bit of stem out of the ground will usually regenerate plants. A horizontal piece, such as stem and leaves, may root as well, but this is not as guaranteed or reliable as the vertical division method.
As far as rapid propagation goes, there are two little known methods (for home gardeners) that can be used to maximize propagation of plants. You don't need to wait for the plants to decide to produce offsets on their own schedules:
1) Take young offsets from the ends of healthy stolons as close to the base of the offset as possible. The healthy stolons tend to replace young chicks taken this way, and will often produce a cluster of offsets at the end of the stolon. Younger offsets take a littler longer to establish and size up, but you get a lot more of them this way.
2) Take vertical divisions as described above. In many cases the plant from the division may rapidly produce multiple offsets as it recovers from the division.
