I've read with fascination Bernhard's many posts on this subject, but I want to ask, if there is no such thing as finger independence, what should we call it when we experience something that feels like finger independence?
Walter Ramsey
There is no finger independence as far as finger "lifting" is concerned. Bringing the finger down, is completely different because the finger flexors do not share tendons.
So one must learn how to lift the fingers by rotation, forearm lifting, etc. You will lift the fingers but as the final and minimal motion of a chain of co-ordinated motions that start at the shoulder girdle.
A karate master uses his fist, but does not punch with it. The punch starts at the hip level and it is transmitted to the fist as its ultimate point of contact. People who are ignorant on the true baiscs of the matial arts start at the fist and concentrate on the fist (make the fist stronger! condition your fists!), sometimes they concentrate on the arms (1000 push ups a day!). But true martial artists know that all of this is silly and ultimately misguided. Instead of worrying about fist conditioning and arm strength, they develop deadly punches by aligning their joints and practising specific co-ordinations of movement that have the hip as its main power-generator (of course for this to happen the legs must provide good anchorage - hece the emphasis on the several leg postures in good martial art schools) - and by targetting vital points.
So, to answer your question, what we feel as "independence" is actually the very opposite: it is
co-ordination between the several elements of the playing apparatus. Faulty technique is always the result of a less than perfect co-ordination (which includes joint alignment).
Unless this problem is solved (and it will certainly not be solved by addressing it as "finger independence" - since then the efforts will be directed at the wrong diagnosis), suggestion of this or that exercise are not going to solve anything. For if you have faulty co-ordination, that is how you are going to play the exercise (or piece): with faulty co-ordination. You may even practise so much that the faulty co-ordination becames comfortable (to a point), and then you will convince yourself of the usefulness of the exercise: "Thanks to [exercise of your choice] I have acquired the technique!" when all you did was to get good at the wrong thing.
I have recently been watching a number of videos by 8 - 10 year olds in youtube. Some of the children are playing reasonablçy difficult pieces, with good musicality and convincing interpretations. But it is painful to watch how they can manage that in spite of some or the most atrociously unco-ordinated motions. Which again goes to show that you can play well (soundwise) with terrible technique. You can see where the wrong motions are all originating from: the main culprit if the lifting high of finger and immobile arms. Then you have - at the other end of the scale - the unnecessary arm motions and swaying, motions of such needlessness and unnaturalness, that it is obvius that they are either imitating their teacher´s way of playing, or have been taught to do so in the name of "beautiful arm choreography" or displaying emotion. One particular 8 year old girl already has so many "use-of-the body" problems, that I wonder if she is not already injured (she certainly seemed to be in pain).
I have also had the opprotunity to watch Lang Lang playing Liebestraumm, and I had to burst out laughing. That guy is a total clown. And he has terrible technique! He made a total pastiche of the piece.

Fortunately I also came across Koji Atwood´s playing the Liadov preludes. What a wonderful technique in its true sense of the word. No wasted motions, the most difficult fingerwork done with ease and comfort , perfect posture, minimum expenditure of energy for maximum musical effects. And to top it all, Koji exudes musical integrity!

And if you want to see more good examples of what is meant by good technique, see (again on youtube) a young Nelson Freire playing Moskowsky etudes, and Sokolow playing Couperin´s "Le Tic Toc Chok" (although his posture cound be improved).
(Mind you, just like you can generate a beautiful sound with terrible technique, you can also have a very good technique -in the sense above - with mediocre musical results.)
Chopinfan´s teacher seems to get the right idea: S/he seems to be investigating the causes of the problem by observing carefully what is going on. Of course, Hanon is not necessary for that. Any sequence of notes will do.
Finally to Chopinfan: Realise that your problem is not muscular and will not be solved by addressing it as such. Your problem is nervous (that is, nerve related). Investigate how you move a finger: Your brain must send a message through your nerves to the muscles that move your fingers. What I am telling you is that you must work on the brain-nerve section of this path. Not on the muscle part. The whole secret of co-ordinated playing (which lead to the illusion of independence) is based on the ability to inhibit nerve messages where they need to be inhibited and stimulating them where thy need to be stimulated. This is slow work at first, but soon it gets programmed into your unconscious and you will do it intuitively, without having to pay any attention to it. But this will come later. For the moment you will really need to focus on it.
Best wishes,
Bernhard.