I'd make a distinction here between management and supervision. The model you describe, working down and knowing the jobs of the techs, is more supervisory in nature. The person in charge has the same technical skills as the staff so he/she can step in when needed, or take on the more sensitive or complicated assignments. This is typically true of entry level management positions in which more direct supervision than management is required. It's also more prevalent among smaller companies having smaller IT staffs, in which everyone by necessity has to get some dirt under their fingernails.
As one advances up the ranks this approach actually becomes detrimental to individual and team success. The purpose of a manager is to manage, to ensure that tactical solutions are consistent with strategic imperatives. A manager who's busy with hands on work can't have his/her eye fully on the ball, and while such a manager can tap out a string of base hits it's a lot harder to put a few over the fence.
Keeping technical skill sets current is indeed a critical competency. One must balance this against where one is and wishes to be in the hierarchy. An IT professional who wants to remain hands on, someone with no interest in project or people management and wanting to stay a technologist, needs to keep up on all the tactical level detail. An IT pro who is or aspires to be a manager needs a different skill set, one that includes strategic level technology deployment and an understanding of the language of business. This is a broader competency, and as such allows one more opportunity to become adept at competency number 10, being adaptable.
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