I never had final exams in high school until I took Advanced Placement courses, and once I started having to take them (high school through grad school), I found them useful in forcing me to review information from earlier in the course, increasing subsequent memory by dint of repeated and spaced practice, but also increasing my ability to see inter-relationships between different parts of the course. Until I took my first finals, I rarely made connections between information taught early on and information taught later. In my subsequent career as a professor, I often tried to structure my courses to have the final unit be one in which all the earlier material in the course could be brought together. For example, when I taught intro psych, I taught about the brain early in the course, then perception, learning, cognition, emotion & motivation, personality, and ended with a unit on the various disorders and therapies. When we covered the disorders and therapies, we naturally reviewed most of the preceding course material, showing how it applied in a new context. I thus found finals pedagogically useful, both as a student as a teacher (and also across fields, when I changed careers from psychology to accounting...the CPA exams are like a series of final exams). But I do think that systems like the old UCLA one put too much emphasis on one exam, and I weighted my own finals to no more than 30% of the grade, and also, particularly in upper level courses, moved to favor take-home exams or final projects which specifically had a focus on integrating the material from earlier in the course. I found that quote about unconsciously conveying the idea that one is "finished" with the material to be just the opposite of my own experience.

Alan Reifman Ph.D.
Are College Final Exams Disappearing?
End-of-semester tests, once ubiquitous, may be fading away
Posted May 01, 2011
Visit a college campus at the right time in May (or December) and you're likely to see throngs of students converging on the library, others racing across the grounds carrying their "blue books" (for writing essay exams) and computer-scorable forms (for multiple-choice), and still others sitting on the floor in groups outside classrooms quizzing each other on concepts from a course they're about to complete.
What I'm referring to, of course, is final-exam week. For my roughly 30 years in higher education, from the beginning of my undergraduate studies at UCLA in Fall 1980 to my graduate studies at the University of Michigan to my current position as a professor at Texas Tech (which I've held for the last 14 years), I've known nothing but finals at the end of an academic term. I've thus been at both ends of the final-exam process and, as I sometimes quip, it's better to give than to receive!
During one of my terms at UCLA, I had three final exams that were worth 65, 67, and 70 percent of my course grades. (UCLA's use of the quarter system -- three 10-week terms instead of two 15-week semesters -- presumably conributes toward heavily weighted finals, as there is not as much time to give midterm exams in a short quarter.) At Texas Tech, which like most universities is on the semester system, my final exams have tended to be worth 25-30 percent of students' grades, only slightly more than what my midterms are worth.
Surely, students would be in favor of "receiving" final exams (especially of the heavy-weighted variety) less often than they currently do, or even not al all. A more interesting and surprising development is that many professors are opting to "give" less. Triggered in 2010 by a Harvard Magazine article and a Boston Globe piece that cited and expanded upon the Harvard article, national discussion of the value of final exams -- or the lack of same -- is more intense than at any time in recent memory.
Harvard's statistics are pretty clear. So ingrained were finals there that, according to the magazine article, "Until the 1940s, [Dean of Undergraduate Education Jay M.] Harris noted during subsequent discussion, requests to conclude a course without a final examination required a formal vote by the entire [Faculty of Arts and Sciences]." Now, according to recent statistics, only 23% of undergraduate courses (259/1137) and 3% of graduate courses (14 of approximately 500) at Harvard had final exams.
To say a professor is getting rid of his or her final exams can have different meanings. In some cases, a professor might substitute a major paper assignment due at the end of the semester (particularly at the graduate level). In other cases, a professor may continue to rely substantially on in-class exams, just more of them with each carrying a relatively low weight toward the course grade. The latter approach seems to have taken hold at some universities within the state of Arizona. The following quote from the Boston Globe article also summarizes some key issues pertaining to frequent, low-weight testing:
There's nothing magical about finals, [University at Albany's Robert] Bangert-Drowns added. They can be arbitrary and abstract - an inauthentic gauge of what someone knows. Research, by Bangert-Drowns and others, shows that frequent testing is more beneficial. And yet, many still find value in the final exam. It might be stressful, even terrifying, but it has the singular power to force students to go back over material, think critically about what they have read, review hard-to-grasp-topics once more, and even talk about the subject matter with classmates and instructors - all of which enhance learning.
As noted above, I've been moving in the direction of more frequent, lower-weighted tests, along with paper assignments and a small portion of student grades being based on spoken participation. I don't think I'll ever abandon in-class exams entirely, as I've had cases in which students' (apparent) deficiencies in understanding the material became known to me only through exams.
Lurking behind the question of whether or not to give traditional, heavily-weighted final exams is the larger issue of how best to assess student learning, as discussed in some of the aforementioned articles. In that regard, Karl and Karen Schilling offered a unique perspective in a 1994 Chronicle of Higher Education article, namely that final exams connote closure and hinder students' ability to see continuity from one course to the next. Quoting from this article:
To students, our assessment practices may be unwittingly communicating the idea that they are finished with a particular set of ideas and concepts. They will move on in the next semester to discrete new chunks of material, they believe; recollection or connection seems unnecessary.
These authors are not against exams per se; in fact, they advocate that courses have entrance exams to promote continuity of knowledge from course to course.
The above discussion is, of course, based only on experiences at a few institutions. I invite readers who have taught at or attended other universities to use the comments section to let me and other readers know any other schools that seem to be moving away from traditional final examinations.