Your teacher is correct, mine gave me the same advice, and I am glad that he did.
Ways to practice this -------------
1. Keep your hands in normal position (yes you will stretch a bit, but don't strain)
2. Practice SLOW, but, instead of doing the jumps slowly, get used to throwing your hand whenver you want to hit a high D sharp, and do it quickly. Play this with a metranome and gradually increase tempo.
3. Try even bigger jumps - Jump two octaves instead of one, this will make the one octave + jumps much easier.
You will notice (if you watch any video of la campanella) that when the pianists play the jumps, they always throw their hand at the high notes and come back, making it look effortless.
This is the only way to conquer that page with relaxation.
Speaking of, remember that La Campanella has much movement involved in it, but this does not mean you should be loud. The majority of the piece should be played very lightly, with relaxation, ease, so if you are tense, you need to slow down and play softly.
If you have any questions at all, please feel free to e-mail me (kingsha@yahoo.com) or keep posting on this thread.
I am actually still having difficulties mastering a couple of sections.
La Campanella is an incredibly difficult piece, so don't let it get you down, and don't overwork it. In a book by Charles Rosen, he mentions on how pianists often have problems nailing and mastering difficult pieces, and he refers to La Campanella and the Rach 3....
Although it is not as difficult as the Rach 3, to be compared in that manner lets you know that it isn't childsplay.