When students write wrong, teachers drift too: SPTA first teaches the teacher to continue from a student's half-finished reasoning
On-policy distillation trains a student on trajectories the student itself generates, but student prefixes can carry reasoning errors or patterns unfamiliar to the teacher, and the teacher's token-level supervision becomes less reliable as prefixes grow longer. Before distillation begins, SPTA trains the teacher on student prefixes of random lengths, rewarding correct final answers, so the teacher learns to recover from student-induced states rather than imitate the student's reasoning; standard on-policy distillation then proceeds with the distillation objective unchanged. On mathematical reasoning benchmarks, students trained with SPTA average better than under standard on-policy distillation and better than under on-policy distillation with task-level RLVR-adapted teachers.