His biggest advancement came in the form of his mastery in Charms, Transfigurations, the spells covered in the Standard Book of Spells from Grade 1 - 3, Defence Against the Dark Arts, Astronomy and Occlumency. He could easily top any examinations that were given to him in these subjects up to the third year of Hogwarts curriculum. As long as the it was the theoretical examination of Potions and Herbology, Harry would be not have any trouble with them either.
