We consider a cooperative system identification scenario in which an expert agent (teacher) knows a correct, or at least a good, model of the system and aims to assist a learner-agent (student), but cannot directly transfer its knowledge to the student. For example, the teacher's knowledge of the system might be abstract or the teacher and student might be employing different model classes, which renders the teacher's parameters uninformative to the student. In this paper, we propose correctional learning as an approach to the above problem: suppose that in order to assist the student, the teacher can intercept the observations collected from the system and modify them to maximize the amount of information the student receives about the system. We formulate a general solution as an optimization problem, which for a multinomial system instantiates itself as an integer program. Furthermore, we obtain finite-sample results on the improvement that the assistance from the teacher results in (as measured by the reduction in the variance of the estimator) for a binomial system. In numerical experiments, we illustrate the proposed algorithms and verify the theoretical results that have been derived in the paper.