If Spotify has been charging you the full premium subscription price, but you are verified as a student this is the solution I found to fix this:
+ go to www.spotify.com/us/
+ log in
+ click "profile" drop down
+ click account
+ in "account overview" on the right side you'll see your subscription
+ click "manage plan"
+ adjust your plan accordingly
It seems that Spotify does not automatically change your plan even though you get verified as a student.