Real Experiments with Grading Mathematics


[ On an unrelated note I see someone’s been going through and grading my essays. I thank you, whoever you are; I’ll take any stars I can get. And I’m also delighted to be near to my 9,500th page view; I’ll try to find something neat to do for either 9,999 or 10,000, whichever feels like the better number. ]

As a math major I staggered through a yearlong course in Real Analysis. My impression is this is the reaction most math majors have to it, as it’s the course in which you study why it is that Calculus works, so it’s everything that’s baffling about Calculus only moreso. I’d be interested to know what courses math majors consider their most crushingly difficult; I’d think only Abstract Algebra could rival Real Analysis for the position.

While I didn’t fail, I did have to re-take Real Analysis in graduate school, since you can’t go on to many other important courses without mastering it. Remarkably, courses that sound like they should be harder — Complex Analysis, Functional Analysis and their like — often feel easier. Possibly this is because the most important tricks to studying these fields are all introduced in Real Analysis so that by the fourth semester around the techniques are comfortably familiar. Or Functional Analysis really is easier than Real Analysis.

The second time around went quite well, possibly because a class really is easier the second time around (I don’t have the experience in re-taking classes to compare it to) or possibly because I clicked better with the professor, Dr Harry McLaughlin at Rensselaer Polytechnic Institute. Besides giving what I think might be the best homework assignment I ever received, he also used a grading scheme that I really responded to well, and that I’m sorry I haven’t been able to effectively employ when I’ve taught courses.

His concept — I believe he used it for all his classes, but certainly he put it to use in Real Analysis — came from as I remember it his being bored with the routine of grading weekly homeworks and monthly exams and a big final. Instead, students could put together a portfolio, showing their mastery of different parts of the course’s topics. The grade for the course was what he judged your mastery of the subject was, based on the breadth and depth of your portfolio work.

Any slightly different way of running class is a source of anxiety, and he did some steps to keep it from being too terrifying a departure. First is that you could turn in a portfolio for a review as you liked mid-course and he’d say what he felt was missing or inadequate or which needed reworking. I believe his official policy was that you could turn it in as often as you liked for review, though I wonder what he would do for the most grade-grabby students, the ones who wrestle obsessively for every half-point on every assignment, and who might turn in portfolio revisions on an hourly basis. Maybe he had a rule about doing at most one review a week per student or something like that.

The other is that he still gave out homework assignments and offered exams, and if you wanted you could have them graded as in a normal course, with the portfolio grade being what the traditional course grade would be. So if you were just too afraid to try this portfolio scheme you could just pretend the whole thing was one of those odd jokes professors will offer and not worry.

I really liked this system and was sorry I didn’t have the chance to take more courses from him. The course work felt easier, no doubt partly because there was no particular need to do homework at the last minute or cram for an exam, and if you just couldn’t get around to one assignment you didn’t need to fear a specific and immediate grade penalty. Or at least the penalty as you estimated it was something you could make up by thinking about the material and working on a similar breadth of work to the assignments and exams offered.

I regret that I haven’t had the courage to try this system on a course I was teaching, although I have tried a couple of non-traditional grading schemes. I’m always interested in hearing of more, though, in case I do get back into teaching and feel secure enough to try something odd.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

6 thoughts on “Real Experiments with Grading Mathematics”

  1. Sounds really innovative!! Do / did students appreciate that this grading scheme is much closer to the way you are judged in real life by potential employers or clients – compared to exams or homework?
    Could you explain in more detail what such a portfolio in analysis would include? How would the professor check for plagiarism?

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    1. My impression is that the grad students in the course — it was about split between upper-level undergraduates and lower-level grad students, because so many math grad students needed the real analysis training — liked the grading scheme. I think that may have reflected awareness that getting some practice with open-ended assignments would probably pay off for thesis work. I think the undergraduates were less sure about it, probably because of less experience thinking about open-ended assignments.

      I haven’t found my portfolio from the course (although I’ve run across notebooks from other classes) so I can’t say exactly what was in mine. I know that it was a mix of the sorts of proving-problem assignments you might find in any set of homeworks for a class like this: the professor gave out suggested problems, some from the book and some that he just composed, with the variation that if you felt like tackling a different problem that could be fine. I believe he also recommended (and certainly accepted) some essays explaining points rather than traditional-style proofs. I don’t know if he’d have accepted writing something about the historical development of a part of analysis, but my hunch is that he would.

      He was looking for some blend of depth and breadth. I don’t have the experience to tell just how he weighted each part of that, though.

      I don’t know how he checked against plagiarism. It can be hard to tell for a straight proof, but an essay or a sentence-form answer would be easier to check. (Well, not so easy back then, which was before the math students were whispering words about “have you tried this new search engine called Google?” around, but easy today.) I think it’d be roughly similar to the problem of telling whether any math assignment is plagiarized.

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  2. I’m going to pass this info on to a buddy who is starting the 3rd act of his career as an elementary or middle school teacher. He doesn’t know what he’ll be doing yet when the next school year comes around, so maybe he’ll be interested in this if ends up teaching math.

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    1. Please do. Oddly, I have the feeling this might be a grading scheme better suited for elementary school than in middle school; it feels a little more arts-and-crafty than the typical. But whether it’s usable depends on the school’s culture and the teacher’s personality and the group of students and things like that.

      I do mean to also write an essay about an alternate grading scheme which I tried and which, bluntly, failed. But why it failed is a further story so that’s worth my muttering about, I hope.

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      1. My areas were philosophy and religion, but I have also suggested to history educators an innovative idea: Teach the subject matter backwards. Start with where we are now and proceed regressively to explain why we’re at where we are.
        I was born in the aftermath of ‘Nam, but have never studied it in a history class b/c we always ran out of time. I know a boatload, however, about the Messypotatoians and the Boston Tea Thingy.

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