I never thought my 1970’s elementary classroom was a place of innovation, but I now recognize it as my first experience with peer grading. For us, it was the exchanging of spelling tests, where we would mark each other’s papers as the teacher read off the correct answers.
I never minded this kind of peer grading because, at least in grade school, I was a pretty good student and I wasn’t afraid for a classmate to see how I’d done.  At the time, I thought it was a method for the teacher to cut down on her work, but I now recognize that the real benefit was to expose us to the correct answers in real time, a chance to reflect on our results and reinforce the correct answers.
I’ve been thinking about peer grading because it is a favorite topic of techno-utopians, probably because if we’re going to make our courses massive, and we’re going to require some demonstration of learning outside of multiple choice exams, someone is going to have to read all that writing and it isn’t going to be the professor.
No one is a bigger proponent of peer response than me. In my composition courses, every draft of every assignment is read by multiple peers who offer feedback. In creative writing, each student is required to read every other student’s story and write a minimum 350 word response and critique. Doing the math, each semester, my creative writing students write more words of critique than they do of their own fiction.
I do these things because I believe one of the best ways to improve one’s own writing is to reflect on and respond to the writing of others. When I ask my students to critique their colleagues’ work, the main purpose isn’t for the original author to receive actionable feedback – though this sometimes happens – but to force them to look at the problem of writing from the other side of experience, the reader’s.
My hope is that this experience causes them to reflect on their own work, that they may see something in how someone else is solving this particular problem of writing and see it as a possible route for themselves.
I believe in this process because it’s something that works for me. I learn more about writing from reading than anything else. When I was stuck on my novel , I turned to some of my favorites for guidance and ended up “borrowing” the structure from an early John Irving book, The Water Method Man. When I’m struggling on a blog post I read Ta-Nehisi Coates to see how it’s done.
But the idea of peer grading, at least in the disciplines I teach, undergraduate writing, does not compute.
I suppose, if we limited the notion of grading to assigning a score, peer grading could be reasonably accurate, probably within 2/3 of a letter grade tolerance anyway, and better than that if a detailed rubric is used. 
But as anyone who has spent time in the teaching of writing trenches knows, at least from a learning perspective, the least important part of grading is the grade. My primary job is to provide not just a number or letter, but feedback and perspective which will allow the student to take the next step forward on his or her journey.
And frankly, peer graders, even very good writers, don’t know how to do that. They are not qualified for the job.
Grading is an expertise, one I’ve developed over many years by reading, quite literally, tens of thousands of pieces of student writing and one at which I continue to improve with each assignment. 
Perhaps this is elitist or self-important of me, but what’s the point of having a teacher if we aren’t going to make use of the unique set of skills and knowledge that they have and the learners seek to develop?
In my creative writing courses, I often encounter students that are quite obviously more talented than me, and have the ability to go well beyond whatever success I’ve achieved, but even those students don’t know what I know. As much as I want them to discover writing for themselves, I can help them shorten their learning curves by pointing out all the false paths I’ve already mapped, all the ditches I’ve driven into.
I’ve said it before , I’ll say it again: education is process, not a product.
This is as true for me as it is for my students. I cannot imagine being an effective teacher if I have someone else reading and responding to my students’ work. This is the only method I have to discover what they need. To work, the process must come full circle: teacher to student to teacher to student.
Last week, Joshua Kim posited that the future of teaching is in making it a team sport . But the work of good teaching, at least in the relatively small class setting which we all agree is ideal – overall course design, individual assignment design, class period prep, grading, conferencing, mentoring, etc… - can’t actually be disaggregated without losing that important connection between teacher and student.
Really, the course doesn’t even exist until the different participants start to mix and mingle. I’ve lost count of the number of times I’ve been teaching multiple sections of the same course in a semester but had vastly different outcomes for both the students and myself. It’s a chemistry thing. This is why the teacher needs to be present and responsible for every last bit, for good or ill.
People who think peer grading, or god forbid, automated grading  is an acceptable substitute for the exchange that happens when mentoring faculty interact with student work don’t really care about education as anything other than something others might be willing to pay for.
No one tweets my tweets but me.
  I’ve heard that this practice is no longer allowed on student privacy grounds, but I have no idea if this is true or not.
  Though if you think instructors are guilty of grade inflation, wait until you see what peers do for each other.
  The obvious question here is if students were getting value from my feedback when I was less experienced. The answer is “not as much,” and at least early on in my career, it would be “not enough.” What grading is for and how to do it strikes me as an under-discussed aspect of pedagogy during one’s TA-training years.