Reading a sonata for fun

Today began with an 18-year-old student reading through a Beethoven sonata with a question in her mind: Where is the song movement?
Working on the premise that a sonata typically has a sonata allegro movement, a song movement, a dance movement, and then a rondo or other sort of finale, this particular piece had only three, which raised the question of what had happened to the song.
To hear her play the piece searching for the hidden song element was to experience it in a whole new light. She had a wonderful time and so did I.
It made me wonder why we approach sonatas as structurally fixed instead of imagining that they explore subject material at itsmost basic.