Jason's personification of his stammer, Hangman, is another element of his colorful imagination (Millennium Falcon, ghost boy on the lake, etc.).
The personification is sort of haunting. The Hangman makes cruel decisions on his own, has his own commandments, and comes and goes as he pleases. He's almost biblical... When he's temporarily defeated by Ms. de Roo, she beats it using her "white magic". He seems to come at the worst times, and recedes when Jason's with Ms. de Roo. It's like... he knows. All of this is almost comical from our perspective, but from Jason's perspective this really is a haunting figure due to both how dangerous the Hangman can be socially and also probably just how annoying it is to have to change words so often.
We talked in class about how it was hinted at that Jason can't full suppress the hangman. When he is excused by the teacher from doing his vocal presentation, a classmate disappointingly asks if Jason will be doing a presentation the next week. The guys are just waiting for him to slip up and really screw up (rather than a short pause) in public. Later we see the hangman strike again in math class, where Jason has to look stupid in order to not stammer (this again shows how important social status is and how damaging stammering would be). All of this contributes to a sort of looming impending doom that this social situation presents.
The hangman presents a problem that must be solved. What will Jason bring to keep him from the gallows pole?
Also, an interesting fact... The Hangman comes from author David Mitchell's person experiences. "I try to cultivate a conviction which states this: "I may stammer on this word, yes, and I may look like I'm being strangled by an invisible man, but if that makes you uncomfortable, then that's 100% your problem and 0% mine."