Re:
I think that previous posters on this subject have a point fibod, although it may not have been articulated in the manner you request.
The "teaching" of airmanship will begin in the first few flight lessons when the instructor will "demonstrate" important aspects of the skill. e.g. prioritization of tasks, decisions made in flight or on the ground about operational aspects (e.g. weather, traffic, fuel, timing etc.). A thorough grounding in theoretical aspects of aviation, which is important for a good knowledge base for descision making, can also be reinforced by the instructor in the early stages of training.
This "demonstration" of airmanship should continue throughout the trainee's program of flight and theory training.
At various stages throughout the flying training process the instructor may seek to involve the student in the descision making and operational processes. This could be likenend to "directing" the student through the process, where the student takes the action but the instructor can guide or influence the thought process. For example, an exercise on precautionary search and landings can be used to re-inforce contingency planning, situational awareness and other "airmanship" skills by presenting the student with several slightly different scenarios on which to base the lesson.
The "direction" through airmanship scenarios will also continue throughout the training, and will hold particular importance during the mid-stages of PPL navigation training and early in the CPL training phase where the instructor can present actual or simulated scenarios for the student and assess or guide his/her responses.
Finally, "monitoring" of airmanship descisions is an important yet almost superfluous element of an instructors assessment of a students progress. Essentialy the student needs to utilize all available resources and come to a descision or perform an action themselves with minimal instructor influence.
In Australia certain airmanship aspects are written into our Day VFR syllabus for the PPL/CPL trainee. This makes it easy to evaluate these aspects as the student progresses and other items may be introduced at the instructors discretion. Essentially, all descisions and actions that the student makes in his/her flying training can be regarded as an indication of their developing airmanship. The level of airmanship of the stduent needs to be assessed by an appropriate instructor as being adequate before the student will be authorised for certain flights (e.g. first solo nav etc.).
To summarise, I propose that airmanship, whilst essentially being "common sense" as tomahawk1673 suggested, involves many important elements of CRM and the descision making processes. Such skills may be taught formally but in GA flight training are most often taught as a component of the actual flying training phase as I outlined above.
Good airmanship is often the result of previous experience, as well as physical skills and spare mental ability (that ability not required to pilot the aircraft). This can be fostered by presenting a student with realistic scenarios throughout their flying training and ensuring they are given the latitude to follow their descisions through to the outcome in a safe, controlled environment.