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the average undergraduate, excellent classroom teaching is the single most important function (by a large margin) that college faculty perform
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Well we definitely disagree on that.
At many colleges thesis writing or some other capstone experience is a required part of the degree, and usually considered the most important educational experience. In fact, getting to the point where a student is ready to do this could be considered the primary purpose of the class work that goes before.
Remember, the goal of college is to prepare the students to educate themselves in the future. During college there is a transition from high school style education to educated adult style education. High school: classroom teaching is at the core of the experience. Knowledge is meeted out in small quantities carefully planned to match the level and background of the students (which can be assumed to be homogeneous). In educated adult style the faculty may be guides who recommend resources, respond to ideas and questions, and provides support for advanced work. The students vary in their funds of knowledge on many areas that serve as background to the topic of the course. Faculty expect, and assume, that each student will have some areas where they need to play catch up, and the most interesting courses bring together students who have varying strengths in different areas.
This is why many graduate seminars (which good undergrads take), meet once per week, rather than the 5 times per week for high school style. The students are assumed to be able to manage their own time, and spend a great deal of time doing their own reading and research. The class serves to introduce new topics, expand upon the reading, and tackle tough questions from the assignments. But most of the learning takes place in the library, lab, and study groups.
For the sorts of students who go to these highly rated colleges it would be a shame if they never moved beyond high school style education.
On the other hand, the PA, as collegehelp has shown, is highly correlated with a small number of factors that most people would accept as meaningfully related to the kind of education that might be available.
If one insists on ranking colleges, then the PA captures a large measure of pure academic accomplishment of the students and faculty. My problem with ranking colleges is that it assumes that all students are homogeneous in their preferences. Only then could any one set of criteria generate a unique rank that is appropriate for everyone. As soon as you recognize that some factors are more important for some students than for others, then the whole idea of one ranking of colleges loses any rational support.
One would need a ranking for a cheerleader who wants to be on the cheerleading team, but does not want it to dominate her life, major in chemistry and go to medical school, live in a large dorm but have a community feeling among the student body, reasonable access to nightlife, but not in a large city... The student who wants to spend as much time as possible studying classics and comp lit, never gets exercise, does not play sports, and loves poetry slams, might have a totally different set of top colleges. Some grand average ranking might be useless to both students.