"Insanity: doing the same thing over and over again and expecting different results." Albert Einstein's famous quote is unfortunately often forgotten. As piano teachers and piano students we need to constantly remember these wise words.
For teachers: It is our responsibility to tailor our teaching style to the personality of each individual student. The younger the student, the more this is necessary. Some students need small bite-sized pieces of information, some do better with bigger concepts. Some need to move around the room more, so do just fine sitting at the keyboard for the duration of their lesson. Not every method book works for every child. Chordal, middle C, intervallic - we need to be well-versed in all approaches and be able to assess our students to decide which method suits which child. More importantly, we need to admit if our initial assessment was wrong and fix it. It's ok to change methods. Don't force a square peg into a round hole.
As the students get older, their personalities are more defined, so that it is easier for us to address a teaching style which will be more successful. They are also more adaptable to a variety of teaching styles, so that we can expect them to try different approaches. This is an important aspect of teaching college level students. They need to see a wide variety of learning possibilities and environments, so that they too can become adaptable when they become teachers. Nevertheless, professors need to be aware of the personalities of their students as well.
For students: I am often reminded of the insanity quote when I listen to students practicing. One thing I like to do is to tell a student that there is "something wrong" in a certain measure/passage and that they must figure out what it is. So I have them play it through again, often slowly, to see if they catch it. More often than not, the mistake is still there. So they do it again and again and again...hoping that the problem will magically disappear. So then I ask them to change "something". This often results in them actually catching the mistake, as they are now focused on what can possibly be changed. Dynamics, articulations, accidentals, rhythm are all things that are often wrong and they know it. This approach allows the student to fix their mistakes by actually understanding what the problem is. Telling students their mistakes is fixing the effect. Having them figure it out often fixes the cause. Students, when you are practicing, think about this. Don't simply play a difficult passage over and over, assuming that repetition will solve the problem. While slow practice is often a good thing to do, it not necessarily the solution to every problem. Doing something slowly will involve different movements and angles, which might have to be changed at higher speeds. Analyze carefully what you are doing - how is the alignment of your body, how about the elbows, fingering, etc. Then change something - anything - and think about the results. What happens if you lean in a bit, hold your elbows out a little, crouch, etc.? You will find that every change you make will have some sort of effect on your playing. And while it might not be the desired effect in that moment, you start to learn about yourself, your body, and what all is possible. You start to become your own best teacher.