Hello Klaus and welcome to Piano Society. I don't think that is 'syncopated' pedaling. It's basically just changing the pedal on each chord. I play this piece with pedal and have never tried it without. Might be a nice exercise doing it that way, though. Have you looked at some of the first editions to see if they show pedaling marks? Back to the fingering on the chords - if your teacher wants you to play this without the pedal, then he or she should help you with the fingering.