There were many famous chess collaborations between the student and teacher. Arturo Pomar and Alekhine, Karpov and Semyon Furman,Kasparov and Botvinnik, finally Magnus Carlsen and Kasparov, Carlsen and Simen Agdestein. Some worked, some didn't.
For those who did work, the chief success factor was on the personality of the student and his view of the trainer. Chemistry.
If we studied the influence of Furman on his pupil Karpov, both men had similar styles, positional wizards to be exact. However, Furman was able to impart his wide body of chess middlegame knowledge to his student. Botvinnik managed to convince Kasparov on the need for chess research despite having enormous talent. He often scolded Kasparov for his impulsiveness in churning out variations without careful study of the requirements of the position. "You'll never be a good player if you let the variations control you instead of you controlling the variations!" I believe Kasparov learnt his lesson under the hands of Karpov in their 2 matches. So Kasparov was a product of the Botvinnik school where deep study and research of the game is the main training method of choice.Botvinnik and Kasparov parted ways mainly due to different political beliefs, but respect is always shown to the master.
Comes the Norwegian wonderboy Magnus Carlsen with his trainer Agdestein. Agdestein believed that the boy learned well when least pressured, stimulated by his own creativity over the board rather than sticking to a fix routine. When Carlsen realised that Kasparov was ready to unleash the Botvinnik approach on his training (which is totally against his learning approach), he had to terminate their relationship.
There were also chief differences in the character make-up in both men: Carlsen, with a happy childhood, always believing in the world of plenty, seemed more like a Mozart compared to Kasparov who lost his father at a young age, often paranoiac of help from others except from trusted sources. Kasparov was probably much more hungry for success as compared to Carlsen, who did not seem to mind if he became the youngest ever World Champion beating Kasparov's record. My opinion is that Carlsen would prefer to stay out of Kasparov's shadow by not going into the record-book race. He will ascend into the Olympus of chess but at his own bidding.
Therefore as coaches, we need to be mindful about the character of our charges and often make adjustments to maximise their potential. Sometimes, that may mean changing some paradigms. No approach fits all. Imposing one's will on a student may create resentment and indifference.