A few years ago I read a book with a young boy as the main character. In this book everybody had magic stones, the boy's stone was small, just a pebble really, he used it for fire magic, and eventually it was broken.
He finds evil shadows lurking in the alleyways of the city, he gets attacked by one which freezes him and gives him hypothermia and he gets saved by an old wizard who becomes his advisor and friend. He goes on some sort of quest to save the sentient during magic of his city, and witnesses a new sinister magic being born that later tries to destroy the magic of his city.
The boy finds a new rock. I remember it was the size of a baby's head, but it was hard for him to use since it wasn't technically his. He saves his city and the two magics protect different sections of the city. The book had a title that was related to fire and the cover was also fiery.