@pfish and danielb:

you too have been very patient with composer, I guessed I lost mine while working with children LOL! I learned a lot from your discussion.

Pfish said:

The actual method is based on the idea that once into it you will be working on 3 pieces at all times. 1 that you are polishing, 1 that you are in the middle of learning and the third piece you are preparing to learn by working on the complicated or tricky parts only.

Yes, that's what we do, too.

Pfish said:

"So they focus on graded repertoire which they first hear on prerecorded audio tracks."  Absolutely, based on the mother tongue theory. How did you first learn to speak? Did you read a speach theory book? No, you were ear trained by your parents and you worked at it till you spoke your first words. So it is the same with any instrument, repetitive listening to the tune played correctly imprints the notes in your brain. Then when you play the tune you automatically want too search for the right notes. It works great, part of the core learning thats proven to work. When we start a new song at a lesson we listen to it together a few times even.

Exactly!! I'd like to add some examples here to support pfish:

One of my nieces when she first started to talk she could not pronounce the G sound. No matter how we taught her, she just couldn't do it. After almost four years, one day, all of a sudden, she got it. 

My daughter (if you read my earlier posts two years ago you might already know this) could not sing in tune, not even twinkle twinkle. I have been singing to her even before birth and playing classical music to her since her birth, nothing help. She has started to take violin lessons since she was 5 1/2 years old along with exploring her own voice in two other singing/music classes and now she sings in tune and could correct me if I play violin out of tune and adjust her own fingers if she plays out of tune.

What I am trying to say is language acquirement and playing/singing in tune is developmental. You have to be exposed enough (that's how we learn to speak in any language) to be able to reproduce the sound right; other could tell you you are out of tune, you did not pronounce that sound right, but no one could teach you how to do it right.

Pfish said:

As far as graded repertoire, children like what they can do. Accomplishment and praise is probably their single greatest motivator. Educated types call this positive reinforcement. Start easy and build skills to more complicated things. Then go back to the easy songs and apply the more complicated skills to develop creativity and improvisation.

This is so true! Last week my daughter and her fellow violin students were performing at a park during Earth Day Festival. After they were all done, audience requested encore but they weren't prepared for that. One boy, if you knew him, you would never believe that he would do that -- he volunteered to play the theme from Harry Potter! He is the shyest in the group, always not wanting to participate in a recital, nervous and feeling insecure when he plays, but he did it because of the request which is the greatest motivation!

Pfish said:
I truley think you are stuck on a "only one truley effective way to learn" mindset. I have to dissagree. I do think we all have the same/similar destination that we strive for...ie musical happiness with our own playing regardless of level or ability. I dont think we are all going to get there the same way, and your way seems most complicated.

Strongly agree with Pfish! @ composer: everyone learns differently: I like to have a teacher who could help me move forward, point out to me when I play out of tune, answer my questions about music and violin, etc, some learn better in groups .... Composer sounds like a very advance player already but he/she is stuck with the idea of what is one truly effective way to learn. @Composer: in mathematics, some of the theory has been proven, but they are quite a few that are just assumptions. If one can not accept that assumption (like me) he/she could not move on. So, composer, I don't know how old you are, but here's mama kim's advice: move on, don't get stuck! The other thing is, when you work on your Ph. D. your questions are all great research topics!