Maybe you don't need to look farther for that "third" method than either Jimi Hendrix or Guthrie Govan.  The way Jimi held his guitar (a right-hander upside down) made traditional approaches moot and he never did music theory.  He just wrangled out whatever sounds he was looking for however he could.  Guthrie started when he was like three years old and he would simply hear something on the radio and figure out a way to reproduce the sound (usually different than "normal" because his hands were too small in the beginning).  In both cases the instrument became an extension of themselves and any application of theory was derived afterwards (this is very evident in the Jimi Hendrix Experience training DVD that I have where, intuitively, you can tell that Jimi just hammered two frets up from a chord because his fingers were there and it sounded good, but the narrator tries to explain how he went to this weird 6th note - yeah, right!).
 

 
"I later learned that it was (or maybe still is) a common way for parents to teach their children in Africa."
 
My first thought upon reading this is that it seems like it might be a precursive form of "call and response" 

backwoods

drewfx1
To throw a wrench into things: I find that a lot of people seem to learn theory by rote. 

So true. I was working towards this in that other thread.

But has any learned the critical theory behind rote?

craigb
But has any learned the critical theory behind rote?

Precisely  I was also going to mention this witticism in my brilliant argument. 
 
(Keep 'em coming boys. I might have a pretty fair hypothesis soon at this rate.)
 
Back on topic. I think the value of rote learning is unappreciated somewhat nowadays. Learning piano, I was made to play the Hanon exercises over and over and over and in different keys too. Hanon is frowned upon by many now because it is seen as mechanical and not following current conventional wisdom about the physiology of the hand. Many teachers say it can cause RSI (but not blindness) but I myself never had a problem with it- you just get in a sort of a trance and your technique gets better when your'e not watching almost.

 
I was reading in the paper the other day the same sort of complaint laid at the NZ education system- they are now thinking that the swing toward "self learning" has gone to far and some teachers are talking about instilling more basic knowledge into kids by way of rote learning. I'll see if I can find that article...

I'm waiting until we can learn Matrix style.
 

spacey
1. Rote
2. Critical

1. Rote
2. Critical
3. Creative

Studing music is a mix of several learning styles:
Below simplified how learning works.
 

1. First you learn the 7 basic scales. (Dorian, Frygian, Lydian, etc) Learn to play them on your instrument. (fingering) This is rote learning.
2. If you study these scales you will learn that there is a system behind the scales. This involves critical thinking. Now can play nicely within key.
3. Third step is that you experiment. You will learn that a combination of notes in a scale played to  a certain chord, within a certain key can have an effect on you and the listener. So you can convey your emotion. That is creative learning.

 
Then the cycle will begin again, because it frustrates you that you have not learned enough to be creative.
 

1. So you listen to other musicans and learn note by note what they play. (rote learning)
2. You will notice in what context they play these notes and for example that it is in some cases ok to play a blue note or a chromatic scale.  This involves critical thinking.
3. You can apply this yourself and you have acquired new tools to convey your emotion. (creative learning)

Zonno

spacey
1. Rote
2. Critical

1. Rote
2. Critical
3. Creative

1.Rote
2.Critical
3.Symbolism
4. By Ear
5. Mocking
 
That's my list of methods, so far.
 
I do not consider a person listening to music, trying to learn a part ( by ear)
the same method as "mocking". To me "mocking" is a method used by a teacher. Again, I will not
argue that- if you believe them to be the same, Ok.
 
 

Don't forget the most overused method: Abject failure!