Miles Teller and J.K. Simmons in “Whiplash.” (Sony Pictures Classics)

To Sir, With Love, Goodbye, Mr. Chips, Dead Poets Society: Influential teachers have been the stuff of classic films forever. But Terence Fletcher, the conductor of an elite music school’s premier jazz ensemble, owes more to a different vein of old-school movie characters – the drill sergeant in Full Metal Jacket, perhaps, or maybe even Jack Nicholson in The Shining.

Whiplash (rated R) isn’t a horror film – not exactly. But as played with haunting intensity by the familiar character actor J.K. Simmons (“The Wire”), Fletcher’s mentor inspires awe and dread among his pupils: He’s a truly malevolent force – Hurricane Fletcher – who emotionally devastates his students in the name of helping them fulfill their potential. And he clearly loves his work.

Written and directed by first-time feature filmmaker Damien Chazelle, Whiplash was the darling of this year’s Sundance Film Festival and wowed audiences in Toronto back in September. It’s the story of Andrew (Miles Teller), a driven drum student who is intent on becoming one of the all-time greats – his idol is midcentury jazz legend Buddy Rich – and who treasures his admission to the prestigious Shaffer Conservatory as a way of making that dream come true.

Shaffer, we’re told, is the country’s top music school, and Fletcher’s invitation-only ensemble is its esteemed inner circle where the best of the best play, as a prelude to careers in jazz. The imperious conductor trolls the school’s classrooms, looking for new potential members; when he recruits Andrew after an impromptu audition, the student envisions a fast track to jazz history. What he gets instead is nothing short of torture: In his first rehearsal, Fletcher throws a chair at Andrew when he fails to meet the conductor’s seemingly impossible standards. He screams at another player’s face until the student runs crying from the room. And that’s just day one.

Pre-Fletcher, Andrew is already laser-focused on artistic success; he grips his drumsticks hard enough to make his hands bleed, and he pre-emptively breaks up with a new girlfriend (Melissa Benoist of TV’s “Glee”) because he expects down the road she’ll distract him from his practicing. But in seeking the approval of his new teacher, the student enters into something closely akin to a masochistic relationship – with Fletcher’s sadistically abusive behavior amplifying the teen’s worst impulses.

In fact, that one aspect of the film is underdeveloped: Whiplash focuses so intently on Fletcher’s animalistic rehearsal-room savagery that it overlooks a proper exploration of what it is about Andrew that makes him stick around for all that punishment. But if Andrew’s inner demons are only implied, what Chazelle does bring to the screen is still absolutely riveting – especially a performance from Simmons of astonishing power, one that’s sure to earn the actor his first Oscar nomination.

I wouldn’t wish a teacher like Fletcher on my worst classroom enemy. But Whiplash is worth seeking out – and impossible to forget.