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That in itself is disturbing.

Thumb under legato without arm swinging should be comfortable if you are using your hand in an efficient way. I say 'should' - we are all different and it is possible you have some physical barrier to this (e.g a stiff joint) and your body works differently.

What concerns me is a student saying 'this is very painful' and a teacher saying 'ok, no problem'.

I was reading about outdated techniques, like playing exercises with heavy books wedged between arms and sides of the body. I can't see how this would make a player better, but I can see how a good player would manage to do it, while a lesser player would struggle. So perhaps the proponent of the system got better despite the exercise not because of it.

and what is the efficient way?
to me at least thumb over was effective and thumb under was ineffective

give peace a chance. I wish i had a teacher to spring some unusual exercises on me.

OK, I'll do my best. The hand is in a good position on the keyboard (by that I mean, there is a bit of an arch in each finger). The thumb joint slides under the arch made by the hand, with the thumb moving sideways parallel to the keys. The thumb then depresses the key, contacting the key at the side of the tip not by pressing the entire length of the thumb on they key. When it presses, the tip of the thumb is lower than the thumb joint.

The key is getting the hand position right before you start, and then sliding the thumb joint. Most people resist this and do the chicken-elbow thing.

Edit: By thumb joint, I mean the very base of the thumb where it joins the hand.

Here is the bottom line. An good, efficient teacher will be observing his student, and teaching according to what he observes. When he assigns something, he also watches to make sure the student will be going home and practising it right,and when the student comes back the next week, he will see whether what he is doing is on the right track. This only works if the student does what he is told to do, how he is told to do it, for long enough.

No teacher here, especially without seeing a student play, can make that kind of judgment call. If they do see the student play, then they call tell all kinds of things.

Meanwhile playing an instrument is not black and white. For example, when you play scales and arpeggios, your thumb will have to do its thing, and the joints of the rest of your body have to not be locked. On one extreme you have a student who has nothing going on in the wrist or thumb and chicken-wings wildly; at the other you have a perfectly still arm with everything locked in, and a hyper-mobile thumb compensating, ready for future injury. Teachers will have GENERAL rules of thumb. A student reading on the Internet may have the idea that these are rigid rules, and will go, "(gasp), my teacher is not following the rules. I'm being mistaught!" Any particular teacher may be going at things from an opposite angle than the majority, yet end up meeting in the same place. Or the teacher is working a certain way because of what he sees with this particular student.

Take an hypothetical example. Two students: one "chicken wings". The other freezes his arm and everything else and uses only his fingers. If teaching can involve learning to use all the joints, or learning to use the fingers, will these two students be approached the same way? The body works as a whole, from fingertip to feet, but what will be emphasized with these two students? (Btw, I tended toward the second while self-taught).

Another example: In the ABF someone put up a tutorial on thumb motion in scales. The teacher was doing the same old-fashioned thing I had learned from an old book; swinging his thumb under the 3rd or 4th finger, then moving his hand. It's what had come close to injuring me. I saw from his demo that I had practised it "perfectly" according to the book. But I also saw small micro-movements in his wrist and arm which corresponded to what I have learned about all the joints being free to move - and this is something that was not in the book I had used. If a ** good ** teacher is teaching along this avenue, he will also be seeing that his student is locking his wrist or elbow or shoulder and will be correcting that. It's not the "teaching method" as in a formula, but how the teacher teaches from day to day. It's not as simple as a student seeing on the Internet how "it's supposed to be taught" and then seeing whether his own teacher is doing this "supposed to be", though we can get a general idea of things.

The big thing is the fact that you talk about pain, with your teacher seeming to shrug it off. There will be soreness from time to time when you do something new that your body isn't used to, but pain is a warning sign. The world is full of people who have been injured due to teaching methods, or following instructions faultily, with a non-observant teacher going by some formula. This is the part that does not feel right: you telling your teacher of pain and your teacher seeming to shrug it off --- IF that is what is happening.

Addendum: TLT has given a good overview of some key aspects.

and what is the diference between soreness and pain?

Keystring has put it better than I could. Many thanks to him!

Always listen to your body, whether it's fatigue, soreness, or pain. ALL are bad and unnecessary if you are doing things right, and not excessively or incorrectly. If your teacher is unconcerned about this causing you pain then you should search for another teacher, no matter how well they can play.

"...and what is the diference between soreness and pain?"

This is a good question--- a very good thing to think about, and to be aware of for your protection. I will offer my opinion, based on my experience as a student; I'm sure others may see the matter differently. Keep in mind, I'm not a doctor or health practitioner (though I've seen plenty of them at times, for these problems).

The continuum goes from:
[1] a slightly lingering fatigue, the feeling that muscles have been used and exercised, maybe for something unfamiliar; a slight sense that the tendons and skin have been stretched a little bit. You might call this 'soreness,' but I would simply call it the feeling of having exercised. There is no pain as such. Rest overnight is sufficient for recovery.

[2] Then we have real soreness: the feeling that we have overdone it. The fatigue lasts longer, the stretching is uncomfortable, there is an ache which lingers for awhile. There could be a slight degree of swelling, and using the hand or arm for other other ordinary tasks causes noticeable discomfort. This soreness might take a few days of rest to recover from.

[3] Really sore: like #2 above, but worse. The recovery phase takes longer, the discomfort is greater, and this is verging closely on danger. It requires that you stop practicing altogether for at least several days. Some people might treat this bad soreness with ice or NSAIDS for relief--- which is ok for most people (unless a doc tells you not to; for example, if you have an allergy, a stomach ulcer, a conflict with some other medication, etc). But, the real solution is to stop overdoing it and also to correct any problems with technique. If we continue to do the same thing, though, the next step is:

[4] Sharp pain, stiffness, swelling, great discomfort with movement. This may last for more than a few days and require a week or more of rest. The tissues and physical structures are being damaged at this point: tearing, inflammation. If we keep overdoing it at this point, the next stage will require treatment by a doctor.

[5] Pain that doesn't go away with ordinary home treatment and rest for more than two weeks. I hate to even go so far to describe the possibilities, but people do this. We are on the doorstep of tendonitis (a long-lasting and troublesome inflammation of the tendons of the hand and wrist, and the processes in the wrist through which they are conducted), sprains (tearing of ligaments), joint pain (possibly, aggravation of existing mild arthritis, or irritation of the joint capsule), tearing and severe soreness of muscle tissue in the arm. It can be worse: the shoulder, back, and neck can be damaged. Long-term overuse or misuse can have many fruits; all of them are a drag. It is time to see a doctor, who may send you to a physical therapist, if you are lucky, for correction; or treat you with medications or possibly even surgery.

So drazh, the difference lies between practicing or exercising reasonably and feeling good, if maybe a bit tired... maybe, going a little further next time, as you get stronger and more skillful... or going all the way, eventually, ignoring the body's healthy message of pain, to seriously bad news, brought on by mistreating the body.

If you are hurting--- and we all know what that is--- stop. If your teacher says, "Oh it's ok," you say, "No, it's not ok, it hurts."

I think even good teachers are not good for all students.

Some teachers play extremely well. Extremely well. Do you think a teacher who was a child prodigy necessarily knows how to teach well? They learned so much when very young I think they may not be as good of a teacher as someone who was mediocre but worked hard in college.

People learn differently as well. If a teacher teachers/learns very differntly from you, then it is much slower to teach you. Teacher may be good, and you may be a good student, but the combination of the 2 of you may be a mismatch.

thank you jeff clef
maybe you mean muscle fatigue for good soreness
because of muscle oveuse
then rest is the best treatment
but because there is no musclr in our finger(only tendons and joint)any type of discomfort (as morodiene said)will be bad and should be avoided

Jeff, just speaking for myself here - I wouldn't be comfortable with any levels on your continuum except [1]. And I'm not even sure I'd be all that happy about [1] either. I certainly would stop and rest if I felt it.

"...Jeff, just speaking for myself here - I wouldn't be comfortable with any levels on your continuum except [1]...."

My exact point.

Well... that, and that it worries me that the teacher is trying to bully the student into doing something that could be physically harmful--- and that the student feels reluctant to stand up for himself. If I'm reading this right. Teachers are supposed to push us beyond our limitations, but there's a point when you have to say, "No...." Or, "Not now."

I agree. The "no pain, no gain" mentality does not apply to something that requires less large muscle strength and more small muscle finesse and arm weight. Not all great pianists are great teachers.

Does "my weak fingers" mean, the fingers on the weak side of the hand (4 and 5) or, all fingers are weaker than optimum?

If I remember correctly, Sandor wrote to the effect that "the fingers should be independent...from one another. Not from the hand."

Opus Maximus -So was I taught to lift fingers high. But were you taught to hold down the idle fingers? If I interpret the OP correctly, that is what his teacher recommended. It sounds very dangerous.