I think you mixed up The Magicians by Lev Grossman and The Rithmatist by Brandon Sandersonas @Radhil mentioned.
The book is about a boy who attends a magic school.
Both books have magic schools as a major part of their plot. And both have a boy as the main protagonist.
There are contests between the groups of students similar to a chess game using magic.
Both books feature magical contests between students.
- In The Rithmatist this is as practice doing their chalk magic, and as a final contest at the end.
- The Magicians features Welters, a game somewhat like chess where a board of squares is conquered via usage of magic.
The good teacher is beaten by the evil teacher in a magic match in a classroom. There's a dark scene of an evil presence visiting the classroom.
- The Rithmatist features one of the lead instructors being replaced via contest towards the beginning.
- The Magicians has a scene where 'the beast' is accidentally relocated into a classroom, and he harms a teacher and students.
They use a lot of chalk to perform their magic and I think the boy's father is a chalk maker.
This happens and I think is only in The Rithmatist
Near the end the group of kids are turned into geese and fly to the South Pole to perform their final test.
This happens and I think is only in The Magicians