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You cannot escape direct qualitative feedback. Even in ‘outcomes based education’, ‘alternative’ programs like the Baccalaureate system, the practice of giving feedback will remain unchanged. However, there are systems that do not use percentage grades, but they are few and far between. ‘Levels’ is one such method.
The issue that arose in my state, Western Australia, is that the practice of portfolios, projects, assignments that had a variety of parts that required assessing… became unwieldy and was not supported by curriculum models prior to implementation. In short, the work of a teacher tripled as the work of the student doubled.
You must also realise - people are familiar with percentages. People find it comfortable to say they’re ‘76%’ when all it really means is that they’re one up from ‘75%’ and one lower than ‘78%’. Cutoffs must happen somehow.
What has happened for some courses however, are required practicals. Presentation of work like art portfolios and samples of work, that show skill and not just a grade. Interviews are mandatory at the university I attended. Many medical schools now require interviews to sift out who are just ‘good at hitting the books’ as opposed to ‘both a hard working student and passionately dedicated and interested in becoming a medico’.
I think this may become the way forward for quite a few institutions, although it does add additional administration to what is usually just ‘fill a quota for a subject’. Another thing that you have to consider - people have to be paid to do the interviewing.

2. Does this mean we have to resort to having teachers pick the valedictorian? (Oh noes!)

In my country, we don’t really have that system… but I do recall that the ‘class prefect’ was a young woman who was not the top Chem student but the most passionate about making it part of her future career. She was responsible, passionate, sensible and worked for what she got. I, personally, would propose to whatever formal body exists at your school to have such a consideration go on before any particular student is chosen to ’speak for’ the cohort?

3. Is it feasible to instill motivation to understand? This would have to be a ground-up change — start with young kids — and parents would have to evaluate qualitative report cards similarly to quantitative ones.

You might be interested in my work on this - I have a series of posts on Skeptical Books for Children that investigates early education on my site.
As for the point about parents - you also have to realise that they have been brought up under a system that used percentages. I have had for the past ten years, parents who came into parent/teacher interviews saying ‘yeah, you said this, that and the other … but WHERE does she come in the class compared to the rest?’ Some ask for percentages. Some ask whether their child is in the ‘top ten’. You can, as I did, give practical strategies and outline where you were heading for the rest of the year with the lessons… but the ‘quant’ does have a bigger impact than the ‘qual’. This is the system that has been fostered for many, many years.

If you’re interested in alternative methods of education, I can tell you some more - but honestly, I’d be (as a student) talking to my student representatives to bring something to the teachers in regards to how you are graded and whether more feedback can be brought to bear in report cards? Or if you think there’s not enough on the work handed back to you, encourage more class discussions on the cohort’s progress so it’s not so much ‘I’m number 7 in a class of 24′.

Podblack.