Interesting thread.
Personally, I think the old adage "a little bit of everything..." is quite a good analogy.
Myself, I enjoy working through the HDG, track, wind and drift triangle on paper, (have a quick method), then using the whizzwheel to see if it comes up with a similar answer. However, I accept that this probably wouldn't work too well in the air! I do like lines on the map, however, because it aids your eyes in looking at where you should be/are going quicker than if you didn't draw one on.
Personally, I put the wind and heading on the map, and other items on a bit of paper.
I also think that Navbox is a superb piece of software, and do not hesitate to use it if I want to, but not always.
Ditto for GPS. However, I learnt my lesson a long time ago about relying on it too much, when I lost the signal perilously close to controlled airspace, and hadn't been concentrating too much on other methods of navigation... A lesson well learned, and one which <hopefully> I won't make again.
Our cessnas are not fitted with GPS, but they do have VOR, ADF and DME, which I see absolutely nothing wrong with using when required.
Talking of GPS, my one BIG failure these days is relying on it too much in my car, but that's another story.
What I'd advocate, is to teach the basics, but not get snowed under with them, and also teach the newer stuff, such as GPS, and flight planning software, such as Navbox.
I think the more methods a person can call on, the better. What works for one person might not work for another, so it's good to have a choice.
It does appear that in aviation sometimes, when weighing up more traditional methods against newer technology, retroverted snobbery is occasionally at work. I'm as guilty as anyone regarding my last statement; What I'd like to be is a brilliant navigator, not needing to rely on anything modern, but getting by, thank you very much, on the good old basics.
BUT I'm not, unfortunately, and am not ashamed to say that sometimes, I need/am grateful for, the helping hand of VOR, ADF, DME and dare I say it, GPS
ariel