On any stringed instument the technique is lifting off not pushing on. Your hand has a natural curl, which you have mentioned, and it is this curl that we need to use. In this curl your finger joints are in line, so tension and pressure on the joints are in line and in correct order. If this is wrong then last joint in the certain fingers collapse, though not a problem to begin with, this will become one.
Though there are many who play flat fingered it is anatomicaly weak as a rule. You use the hand to push the fingers on, the fingers are so straight the presure is in the wrong part of the joint and out of sequence, it is the main knuckle that is controlling it.
To illustrate this for you, take your fingers straight and put the tips on the edge of a table and push down a few times and notice what is going on and what you are feeling. Now curl them and do the same, taking time to notice and feel. In the straight finger you will see that the main knuckle takes the strain and the fingerjoints collapse back. With curled fingers the pressure is spread even and the tips of the finger have a better support because the are now all level, another factor of finger curl.
So how do you keep your fingers closer to the fretboard? You learn and understand that it is that curled finger lifting off that need to be educated.
The exercises you need to do are about opening and relaxing not about gripping, let me explain, squeezing exercises, build a powerful grip, so if you squeeze say a squash or tennis ball or one of those hand exercisers your building the wrong muscle movement i.e. a gripping action. This is the same principal when you play a lot of bass the wrong way, you are in effect developing a squeezing action, that is pushing(squeezing) the fingers against the srings using the neck as resistance.
Put your hands on the bass, now your left hand lifts the fingers away from the neck, you relax the hand and the fingers come back, You then use minimal muscle tention to hold it away from the sting o apply a little bit more to make it fret. Try this, put your hands flat on a table and lift each finger off in turn, the thumb is easy, as is the forefinger, but you may find the others a bit tricky as the finger next to it will get involved. Practice this till you can do it with out thinking, any finger at any time you can lift up at will, to any height, this reinforces the fact your using you hand correctly, so do this exercise for five or ten minutes ever day or incorporate it in to your warm up. If you have weak hands to begin with a hand exerciser will help get you to a better position to work from but only to get you there. You should find that the act of playing and daily life will keep your hands in condition if looked after properly.
Now as you play it becomes about how far from the fretboard and strings you lift your fingers not about putting them on, and its reasonable to say if you don't lift them that far, they don't have far to come back, so speed and fluency will follow, the same applies to the right hand.
Remember that with no effort the hand will curl, that is it's natural relaxed neutral position if you will. straighting the fingers than relax and they will return to the natural curl with no real help. The same is the other way close them, then relax and they will return to the natural curl. So in fretting it is lift relax, fret relax, lift relax, fret relax, and so on with the fingers.
Not lift off put on lift off put on etc, because that is developing a grip.
Congratulation to you teacher for informing you to do something about it, it is an often over look point, you will reap the benifits in later life.
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