In her new film “The Kindergarten Teacher,” Maggie Gyllenhaal plays a frustrated aspiring poet who discovers that a boy in her kindergarten class may be a budding literary genius, and begins co-opting his verses as her own.

When Gyllenhaal was preparing for the role, she thought a lot about what sort of poetry her character, a Staten Island teacher named Lisa Spinelli, would write. She figured Lisa’s poetry would be somewhat labored and clichéd — maybe verses about flowers and butterflies. So she and the film’s writer and director, Sara Colangelo, decided to ask a real poet to write some lines for the movie.

Commissioning poems wasn’t easy, it turns out.

The first verses they solicited, from the poet Dominique Townsend, an old friend of Gyllenhaal’s, weren’t quite plausible. They were too layered and complex — too good.

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