In the process, Mr. Deak said, the students have to become leaders: “Look at a 10-year-old who comes up to a bassoonist’s kneecaps and says ‘That’s too fast’ or ‘There’s something wrong with that note.’ They have to defend their pieces, and boy, do they do it.”

Indeed, Ms. Cowan and Ms. Millar did exactly that in rehearsal with the Philharmonic.

“I was a little scared, because I felt like they already knew what they were doing,” Ms. Cowan said. “I told them, ‘Can you make this a little faster?’ But overall, I think the piece was played perfectly.”

In the Very Young Composers classroom, the environment is broadly supportive as the students share their latest works. Still, Mr. Deak said, “some kids write pieces that have legs, that really take off.” These two girls, he added, “wrote the hit tunes.”

Outside the program, Ms. Cowan is a violinist and pianist; Ms. Millar plays piano and clarinet, and has sung with the Brooklyn Youth Chorus. At home, though, neither listens to much classical music.