There are several strands of research that I find fascinating with regards to problem solving. first, what are the steps of problem solving. Heather with Jumpstart uses a 6 step approach , that really amounts more to conflict resolution than it does a more measured research approach. There are visual methods - sketching out various solutions. The scientific, classic view of determining alternatives and weighing each one. There are processes used by people in high stress positions where a single option is weighed, searching for a reason why it wouldn't work, and when 1 solution is found it is implemented without ado. Most of these are more based in the process of decision making though- and problem solving and decision making, aren't precisely the same thing. Really, the decision is really only 1 step of the problem solving process. It is also interesting to note that some of the best problem solving techniques I have seen have come out of resources devoted to ethics, and ethical decision making. At any rate, for me the start of problem solving starts with defining the problem, taking stock (gut reaction, what I know, what I know others know, ect...) then taking action steps to gain more information, and so forth. What is important to note here however is that problem solving is a great way to look at double loop learning as your assumptions are very important in determining how you respond to a challenge. For someone with a history of automation, the easiest solution may be a pneumatic cylinder, for someone else, that might be a last resort, simply because of prior experience.
At any rate, it often seems that problem solving visually is often practiced by scenic designers, and often not tended to as much by TDs (though they occasionally do it without thinking about it). This can be done in multiple ways, physically through the manipulation of items (mock ups, tests, models) or by drawings both by hand and in CAD. CAD in particular helps identify cable paths, fleet angles, travel paths and the like. If you draft in 3-d you can see the object from multiple views to check for errors that way. Layers can be adjusted and/or locked to plot only what is needed, while saving your work for future reference. It can be a powerful tool - and one I believe is underused.
While looking for more information on this topic I came across the following website:
It has alot of good stuff. I particularly think that transferring shapes to flat pictorial drawings and back are useful to develop spacial / visual skills. It is ironic that in an age with such computer savvy-ness that spacial relationships are sometimes so hard to teach . Also on the above site if you go to the schedule section and then click on 'electronic reading pack' there are some good pdfs. I think a class like this would be interesting to teach, and if nothing else, could be an good intro to drafting.