This could be because the user or the email client store the Schools notification/email address as that User. The user should be able to change the From Name in their Email Client. 

For example, I add yourschoolname@schoolnotifications.org to my address book as Joe Smith, then the Faculty Member changes to John Smith this will still show Joe Smith because that's how I saved it in my address book, or the Email Client saved it as such. You should be able to go in and change this name in the Email Client.