When Right Meets Wrong: Teaching LLMs by Letting Their Mistakes Talk
Training a reasoning model is often treated like running a classroom with a very impatient teacher: give the model a problem, let it produce several answers, mark each answer right or wrong, and push the policy toward the winners. That is already useful. It is also slightly wasteful. Because in a real classroom, the wrong answers are not just trash to be swept off the floor. They reveal what the student misunderstood. They show which shortcuts are tempting, which algebra step keeps breaking, and which false pattern looks suspiciously persuasive. A good teacher does not only praise the correct solution. A good teacher puts the correct and incorrect attempts side by side and asks: what exactly changed? ...