In what situations would "self-plagiarism" be authorized by a professor?
I first note that the idea that students' work is somehow not a part of their personal body of work, thus owned by them, and re-purpose-able however they like, has always struck me as very strange. In fact, this seems to me quite contrary to any notion that universities are creating intellectuals and scholars. That is, there is a powerful implicit assumption that students would never do anything at all if not commanded to do so by instructors in officially scheduled courses, so somehow do not deserve credit for any work, and do not truly "own" it.
I myself have absolutely no objection to anyone's re-use of work they happened to do before I asked them to do it. :)
All the better, in fact, if they had the sense and drive to do it before being asked.
I much prefer the idea that each student develops a body of expertise, often expressed in write-ups of various technical sorts, if only as notes-to-self, which can be brought to bear in various situations. Not that everyone is required to give up all fruits of their own prior work, to avoid some highly commodified, artifactual stigmas about "self-plagiarization". Sometimes the answer to a question does not change just because it is asked repeatedly.
I have seen advanced classes where a previous paper was substantially revised, with the approval of the instructor. This is likely to be allowed only when the new work is clearly building on the old work, rather than using the old work as a shortcut. That is, it most often is approved because the student has a passion for the topic and wants to dive in much deeper in a subsequent class; it is not generally approved because a student has "already done the work" and will essentially be turning in the same thing twice.
Usually this happens on a case-by-case basis, but I know of at least one advanced writing course where the actual assignment was to revise a paper that had already been written for some previous class. This allowed the professor and students to focus on the mechanics of research-based writing, without having to actually conduct most of the research.
Here are some examples from my Department:
In their second year class on English for Academic Purposes (EAP), students are asked to provide an example of assignments they submitted in their first year. Their task is to evaluate their earlier writing in terms of the quality of academic English not the context or technical content of the work. They are graded on the basis of this evaluation. Students find this especially enlightening because they can see firsthand the improvements they have made in their study of English. For example, they are able to see how much more proficient they are in the skill of citations and bibliography construction.
At the end of the third year, students submit a research proposal for a methods course. This proposal is revisited in the beginning of the next year, serving as the basis for their undergraduate dissertation.
At the start of a course on basic biostatistics, students collect data and pool them. The data are used in four other biostatistics courses over the next three years to demonstrate techniques of increasing complexity.
In each of these situations, we have strict controls applied through the moderation process that prevents the assessment criterion from being applied more than once when the material is re-used. Take, for example, the third scenario above. One of the marking criteria during the data collection period is, say, appropriate construction of a Likert scale question. This criterion does not exist in any assessment task subsequent it. This prevents us from passing along a mark from assessment to assessment over time.