Standard incentive theory talks a lot about the principal-agent problem. The general idea is that you have a manager (principal) and an employee (agent). The principal wants the agent to exert the "correct" level of effort, but the principal can only observe the output of the agent and cannot see directly the level of effort exerted. Furthermore, there is some noise in the process such that the level of effort cannot be recovered from the output. The solution to the problem is generally to offer an incentive scheme that is a function of the agent's output. This serves to better align the objectives of the principal and the agent.
None of this is anything new. BUT, coming back to the teacher incentives issue, there seems to be something other than the standard principal-agent problem at hand. School boards, principals and presumably society want students to perform well. Clearly, students would be the agents in this situation. However, there is not (currently) a direct link between the aforementioned principals and the students. Instead, there are teachers in the middle. Therefore, Greg Mankiw's question about who should be given incentives within this framework is an interesting one. In other words, are teachers agents, principals, or both? Are school boards principals for only the teachers or also directly to students? I believe that it is important to answer these questions before even addressing what teacher outcomes should be rewarded.
In standard incentive theory, an incentive in place is designed to induce effort, not quality, at least in the short-run. (One could argue that in the long-run the incentive would induce quality as well due to employee sorting.) Clearly, effort is what needs to be induced from students. (Even if students need to build human capital/quality, effort is the defining component here since there is no sorting of students.) With teachers, however, is it effort or quality that needs to be induced? In other words, do teachers simply need to try harder, or do they specifically need to build their human capital? This distinction is important when trying to design the appropriate incentive scheme. Furthermore, a reward based on student performance is a largely useless incentive unless teachers know specifically what they can do in order to generate more successful students.
My hunch is that Greg is onto something with the suggestion that both teachers and students should be given incentives. Anecdotally, a lot of the failures in the current academic system are in situations with high-quality teachers and unmotivated students or motivated students with less than hard-working and/or unqualified teachers. If this is true, then neither teacher nor student motivation is sufficient in itself, and there are large complementarities at play. (Full disclosure: it is my strong belief that educators should be exactly that and should not be expected to take on roles of social worker, motivational speaker and the like. There should be a division of labor where other employees perform these more motivational tasks, and the teacher's motivational responsibility is to make the material clear and interesting.) Offering incentives to students (theoretically) will induce effort, especially if this is augmented with non-teacher motivational help. (More on this later.) Offering incentives to teachers will...do something, though no one seems to be sure what. If I were a teacher that was to be compensated on student outcomes, I would offer to share some of my surplus with my students. That said, it's easy to see why these issues are still open to debate.