Your SSN should not change. It is your ID.
When I first came to US, I am on F2, but I got my first SSN card with note 'not authrize to work'. Then I got my F1 and change my SSN card with note 'Valid to work with permit'. It is the same number.
When I first came to US, I am on F2, but I got my first SSN card with note 'not authrize to work'. Then I got my F1 and change my SSN card with note 'Valid to work with permit'. It is the same number.