So, what about grading in that inverted transition-to-proofs course? Other than the midterm and final exams, which were graded pretty much as you might expect, we had four recurring assignments that required grading: Guided Practice, Quizzes, Classwork, and the Proof Portfolio. Let’s discuss the workflow and how it was all managed.

Let’s start with the easy stuff: Quizzes and Guided Practice. Quizzes were done using clickers, so the grading was trivial. Guided Practice was graded on the basis of *completeness and effort only*, on a scale of 0–2. So it was almost instantaneous to grade. Students would submit their work using a Google form that dumped their responses into a spreadsheet. I would just sort the spreadsheet in alphabetical order, look through for any glaring omissions or places where effort was lacking, and then put the grades right into Blackboard. A grade of “0” meant you submitted nothing, or might as well have. A “1” meant that something was missing — an omitted exercise, a response of “I didn’t know how to start on this” (lack of effort), and so forth. A “2” was anything else.

Students did not have to be correct on these — just show evidence that they *tried* to be correct. On Guided Practice it is important that students should feel free to make mistakes, because I need to catch those mistakes to repair them before the Classwork starts. I was afraid that if I graded on correctness, not only would this take a lot of time to grade, students would be more likely just to not try at all than try under the pressure to be right — thereby robbing me of crucial data about student misconceptions at the early stages. Did this inflate the grades? Not really — these Guided Practices were only 5% of the semester grade.

That brings us to Classwork. This was graded on a scale of 0–10 on *mathematical correctness, effort, completeness*, and *writing quality*. The rubric I used was:

**10**= All problems are completely done and submitted on time. There are no mathematical errors in the work, and all solutions are clear and easy to follow.**8**= All problems are completely done and submitted on time, but there’s a minor error (note singular) in writing or in mathematics.**6**= All problems are done on time with a reasonable attempt at completeness and correctness. But at least one problem has one major error, or there are significant errors in writing, or numerous minor errors are present.**4**= Most problems are done on time with a reasonable attempt at completeness and correctness, and all problems show evidence of making an effort to be completed. But some problems are not finished or represent a less-than-complete effort at finishing with correctness. Or, all problems are submitted but contain numerous errors.**2**= Less than half of the problems show evidence of serious effort to be complete or correct.**0**= Essentially no progress has been made towards correct solutions to the problems.

I’d grade these just by sorting them into what I felt was the appropriate “bin” — reserving the right to assign an odd-parity grade if the work was on the borderline — and then give detailed notes in ink on the students’ papers.

Classwork was very easy to grade since I was actively involved with the students as they came up with their solutions. I knew what I would be getting from them in terms of a product. Plus, they were handing in group writeups, which means 4–5 papers to grade per section rather than 16–20. I estimate that took me less than 30 minutes to grade two sections’ worth of Classwork on a single day, on the average.

That brings us to the Proof Portfolio. *That* was challenge.

The Proof Portfolio is the centerpiece of the course and the main instrument used to have the course satisfy our Supplemental Writing Skills requirement. This assignment consisted of ten problems that involved proofs, to be completed by the end of the semester. It was given out during week 2 of the semester. But students didn’t just write something up and hand it in during week 14. A key component of the Portfolio is to engage in multiple drafts and revisions that converge upon a final product, with instructor feedback. So each student had the opportunity to turn in two preliminary drafts of each problem, get feedback and a provisional grade from me on each submission, and then submit a final draft prior to the last day of class.

Do the math on this and you’ll see why it was a challenge. Each student will be handing in 2 preliminary drafts to me and getting feedback on each problem. Times 10 problems, divided by roughly 12 weeks — and we’re talking roughly two draft submissions per week. In fact the requirement in the class was that students would submit two drafts per week, one new draft and one revision of an old draft. (Getting a perfect provisional score on an early draft exempted them from the second weekly submission.)

This schedule implied two things: First, there was no room for taking my time if students need feedback from the first draft to potentially make changes for the second draft that will be submitted later in the week. I had to grade *fast* and there was no procrastinating. Second, I had 40 students — times 2 drafts per week, which comes to **80 proofs to grade each week**. And this was on top of preparing problems for classwork and (crucially for me) making screencasts for the out-of-class viewing. (The screencasts along were taking upwards of 8 hours a week at certain points in the semester.)

We’re verging into the grading load territory of history and English profs here, which is actually sensible because this was, after all, a designated writing course. I should emphasize here that the grading load was *not* due to the inverted nature of the class. All sections of this course have more or less the same grading requirements because of the way the course fits into the whole curriculum.

I’d love to say that I came up with some brilliant shortcut for making the Portfolio grading less onerous, but I didn’t. I just had to woodshed it. A few things that did help were:

- Using a simple rubric to assign provisional grades to each draft, plus the fact that these grades were provisional and didn’t count, hence less pressure to be exactly right about the grading.
- Using electronic tools to grade and annotate the work. Students submitted their work to a specially-created GMail account as PDF’s. I’d open the PDF in Preview, annotate electronically, save the file and attach it to an email and shoot it back to them. By saving the file, I had an archive of students’ submissions and provisional grades (especially handy if a student missed a submission deadline or had a grade dispute). By using electronic submissions, if students were getting the same single thing wrong on a proof, I could create a stamp that I could just cut and paste into their work to save writing time.
- Emeril’s Big Easy Bold to help me get up to grade 5am or stay up to grade until 1am, which I sometimes had to do.

As much as the Portfolio grading was a burden, it helped keep me calibrated with my students. I mentioned before that one benefit of flipping the classroom is to give students in this class more time and ability to focus on the Portfolio. The difficulty of the grading helped me to remember just how much time and effort they had to put into this assignment as well.

And the more I look back on the class, the more I see that inverting the class actually helped me in the grading process as well — as I am more actively involved with my students during their hardest work, I don’t have to work as hard to grade it later.