Newest Grading Scheme

Grade cutoffs
This semester’s grading scheme is based on the one I developed (by stealing ideas from a lot of different people) in Fall 2010. However, I started incorporating more inquiry-based learning in my courses, and so I have adapted this scheme to incorporate the IBL stuff.

Before I describe the system, here are the goals I have for any grading scheme:

  1. I wanted a grading scheme that acknowledges that students learn at different rates. I do not want a student who understands the material well at the end of the semester to be penalized for not having something down in the first month of the semester.
  2. I wanted a grading system that gave specific feedback on how the students can improve. A student who gets every “Find Tangent Lines” question (see below) wrong on a quiz knows better how to improve than a student who gets a 70%.
  3. I wanted a grading system that requires that students demonstrate success over an extended period of time—cramming one night won’t lead to success.
  4. I wanted to create low-stakes grading system. Students are not penalized for getting a question wrong on a quiz; they simply do not help themselves.
  5. I wanted to create a grading system that requires that students learn the material. There is no way around it—students need to learn all of the skills in order to get a decent grade.Then I tried to figure out what a C-student should know, what a B-student should know, and what an A-student should know. I decided that a C-student should be able to do the mechanical skills with some fluency, but not necessarily much else (this is clearly debatable). An A-student should be able to solve challenging problems, and the B-student should be somewhere between those (note: this clearly needs some more thought).With these thoughts in mind, I decided to have a two-tiered grading system. I devised a quiz system to determine whether a student would get at least a C; the remainder of the grading system is based on student presentations of problems and a final exam. For those of you who are into this sort of thing, here is my syllabus, which spells everything out in detail.

    The quiz portion is a standards-based grading system. I chose the following procedures for my calculus I students:

    • Graphing Functions
    • Shifting and Scaling Functions
    • Trigonometry
    • Limits
    • Definition of Derivative
    • Derivatives
    • Tangent Lines
    • Second Derivatives
    • Linear Approximations
    • Extrema
    • Integrals
    • Fundamental Theorem of Calculus

    For my calculus III students:

    • Parametrizing Curves
    • Tangent Plane Approximations
    • Derivatives
    • Lagrange Multipliers
    • Double Integrals
    • Triple Integrals
    • Line Integrals
    • Parametrizing Surfaces
    • Surface Integrals

    Each quiz question is tied to one of these topics (here is a recent quiz). In order to get credit for a quiz question, it has to be completely correct—no sign errors, arithmetic errors, etc. On the other hand, there is no penalty for a wrong answer. I grade these simply by counting how many problems each student gets completely correct during the semester. A student who gets four questions in a topic (eight questions for “Derivatives” and “Integrals” in calculus I) is declared “done” with that topic, and can stop answering questions on that topic. Once a student is “done” with every topic, that student has successfully completed the quiz portion of the class.

    I really like this set up. Many, MANY students who have been struggling all semester are making a lot of progress on the quizzes now (I have been telling the students all semester long that they should expect the graph of their quiz numbers versus time to be concave up). Moreover, the students have to figure out everyone of the processes—they cannot slide by with a 60% on, say, The Fundamental Theorem of Calculus, and make up for it with 90%s in other topics.

    In short, I really like this system because it is both easy on the students (missing any one quiz question does not hurt a student’s grade, unlike the usual “averaging” system) and hard on them (they are expected to gain some fluency with every single topic).

    A student will get at least a C in the course if and only if that student completes the quiz portion of the course. I really believe this should be attainable for almost every student, since these topics are really just a matter of following recipes (the only exceptions are students who have exceptionally poor algebra skills).

    To determine whether a student gets a C, B, or A, stay tuned for next week’s post.

    (Image “Grading cutoffs” by flickr user ragesoss)

12 Responses to “Newest Grading Scheme”

  1. Joss Ives Says:

    Hi Bret,

    I’m currently trying to nail down my grading system for my winter Intro Physics II course and I appreciate you giving me some stuff to chew on.

    As we all know, students seem to find ways to break any well-intentioned system. The most likely situation I can see is a person who does B-level work on the exam and presentations, but does not manage to get 4 of the 8 quiz questions for one of the topics. How would you deal with something along those lines?

    • bretbenesh Says:

      Officially? That student would get a CD (my school’s grade between a C and a D).

      Unofficially? (I hope my students aren’t reading this:). I view all grading schemes as something that guarantees students will some grade, but I reserve the right to give a student a higher grade in certain circumstances. In the traditional grading system, I would still consider giving a B to a student who gets a 78% when the cut-off for a B is 80%. I need to be able to justify it to myself (e.g. The student did really well on presentations and exams), though.

      Another factor with my particular scheme: it depends on how close the tru dent was to completing the quizzes. If the student all 4’s except or one 3, then that student will likely get a B, even though the student did not technically “earn” a B.

      In short, I do not want to be a slave to any grading system. I suppose the main purpose of any grading system is to communicate expectations to students. Beyond that, I am happy to break the system if I have enough evidence to justify the grade.

      How are you thinking about grading Physics II? Bret

      • Joss Ives Says:

        Students, stop reading Bret’s blog! There is nothing for you to see here.

        I’m being very indecisive about my Physics II grading right now. I didn’t teach Physics I this year so all the students are going to be new to me. I want to be doing something SBG-ish, but my course is already quite different from their Physics I experiences so I’m not certain if a completely different approach to grading will help or hurt buy-in.

        I’m really liking your method of binary grading and coarse standards. I need to sit down and crunch the numbers a bit to see if I feel that I would be able to give them sufficient assessment opportunities to make this work for my own situation.

      • bretbenesh Says:

        Hi Joss,

        The main problem: I _invited_ my students to read the blog in the syllabi. But, fortunately for me, I don’t think any of them read it.

        My opinion for you is: grade however you like. The class is one big working whole, and you should choose the grading scheme that is going to complement everything else that you do.

        There is a concern about getting enough assessments in. I didn’t quite do it in multivariable calculus. My solution was to lower the number of questions that are required, so that worked. But it was not ideal.

        In previous semesters, I let students come to my office for one quiz question per day. I stopped doing that—it was a lot of constant work—but I would consider doing it again.

        I am going to need to make sure I can carve more time out for assessments next semester. In calculus I, I had the luxury of two 70 minute labs each week; I used at least one for quizzes. I didn’t have that in multivariable (which is why I ran out), and I won’t have them in probability and statistics next semester.

        Let me know if you think of anything brilliant. Bret

        Let me know

      • Joss Ives Says:

        I have three 80-minute lecture blocks per week. In the past I have had a 30 to 40 minute quiz once per week and if I went to SBG I would dedicate one class a week to quizzes. In fact I might do that either way. My class size is 36 so individual reassessments are tough, but not impossible. I think I have just convinced myself that it is time to sit down, make a nice big pro/con list and decide what is going to work best this coming term.

      • bretbenesh Says:

        I have three 70-minutes lecture blocks per “week” (it is complicated here), and I used one class per week for quizzes in multivariable. I also waited until the fourth week to start quizzes, which was a mistake. Next semester, I will likely use one class per week, but start immediately (perhaps including some “remedial” topics to jog their memories and provide quiz material for the first couple of weeks).

        As I alluded to before, I had FIVE 70-minute lectures blocks per week in calculus, and I used 1-2 of them for quizzes. This was no issue at all.

        So my recommendation is that you get your school to give you one more lecture block per week. Bret

      • Joss Ives Says:

        Realistically I can have as much class time as I want because there are unofficial channels such as tutorials/recitations sections that we don’t use but could. But the flip side is how much time is it realistic to ask students to dedicate to a single course? I find myself constantly battling against acting like my course is the only one taking up their time. I think a median of 8 hours per week hits a sweet spot where they don’t feel frustrated that the course is taking over their life.

      • bretbenesh Says:

        You are such a nice person.

        That is a tough question. I am in the midst of struggling with how to get enough quiz opportunities in my class right now. I have no solutions. Bret

  2. Adam Glesser Says:

    Hi Bret,

    It is very interesting to see your choice of standards compared to mine. We are covering the same material, but I break it up into four times as many standards. Very often, because I break things up so much, I find myself struggling to decide which standards are involved in a problem to a degree worth assessing. I’m curious if you have any sort of problems in the opposite direction, e.g., using the grade book to communicate to a student that is struggling with surface integrals that it is because they don’t understand how to take the cross product of vectors.

    From your syllabus, it sounds like you race through the material so that you can cover it several times. Is this accurate? Is this how you can guarantee so many assessment opportunities on material from later in the course?

    One observation my students made this semester is that my grading system doesn’t penalize them enough for not knowing basic material. For instance, I will give a student a 3.5 (on a 4-point scale) if they can correctly set up a double integral over a reasonably complicated region, use Fubini’s theorem to break it up as an iterated integral, and then biff the landing because they can’t remember how to integrate sin^2(x). Their suggestion is that I have explicit standards for Calc I and Calc II material so that I can give them better feedback about what they need to study, and so that they have an incentive to go back and study it. How do you deal with this problem of assessing material that is supposedly “remedial”?


    **************Substance ends here*************
    Line 21 on page 2 of your syllabus should be “To do this…” rather than, “Do so this…”

    • bretbenesh Says:

      Hi Adam,

      First, your last line is terribly substantial! I often re-use large parts of my syllabus, so your pointing out of that single error actually saved many errors in many different classes!

      Second, I do cover all of the material twice during the semester. We cover the entire semester by mid-semester. As you suggested, one huge advantage is that I can assess every single topic for at least half of the semester (if that makes sense). (The only way it is possible for me to do this is that I flip the classroom in the first half of the semester. They read at home, and we spend more of class time processing the reading).

      Regarding granularity: I don’t know that my way is the best. I struggled with this, though, when I was making up the syllabus. I decided that I really just needed to get a sample of possible topics so that I can tell whether it is likely that a student knew enough to earn a C. This has the advantage of cutting down on testing time—the more topics, the more questions I need on each quiz. This means that I lose more class time than I already am to testing.

      What are your reasons for having more topics?

      Your students don’t think that you punish them enough for basic mistakes? I suggested switching to binary grading—they either have it completely correct, or they still have work to do. Not only does this send the message that they need to know every part of the process (they cannot get by without ever learning the cross product, for instance), it makes grading much, MUCH easier. I combine this with video solutions ( to make grading to much faster without sacrificing much (any?) feedback.

      Finally, I think that it is smart to include some “remedial” standards. I did this in calc I (“Graphing functions,” “Shifting/stretching functions,” “Trigonometry”), and I wish I would have done it in calc III (“Derivatives,” “Integrals”). It encourages students to review, and it also gives you something to quiz them on in the first two weeks of the semester. (That said, it does mean that it takes more time for quizzes).

      You asked a bunch of great questions. What are you thinking about these topics? Bret

  3. Adam Glesser Says:

    Hi Bret:

    I think I’ve always tried to keep my standards extremely focused so that
    a) it helps me write test questions that cover everything I want to cover
    b) the scores represent very specific feedback about where the students are struggling and succeeding.
    For example, I break up “Parametrizing Curves” into three separate standards: Parametrizing line segments; Parametrizing circles and ellipses; Parametrizing explicitly defined functions. This allows me to test give them feedback on the different skills separately. The disadvantage is that I now need to test all of them many times and this is a lot of time spent assessing. The advantage is that the student knows precisely which curves are giving them trouble. Let me use some temporary vocabulary: I’ll call my more granular standards “skills” and your broader standards “superskills”. One thing I’ve been mulling over is creating some concept maps for each superskill in terms of the skills. Perhaps that would be sufficient for helping them see the details, while allowing me to give feedback only on the broader set of superskills. Still, I would hate for them to get the line parametrizations and explicit function parametrizations correct four times, but not ever the circle parametrization.

    The idea of compressing the course down into half the semester and then running through it twice is really intriguing to me. I’d given it some thought in the past and decided it was too radical a departure for me, but hearing your description is making me reconsider. I suspect it is easier to do with a procedural class like calculus, so I might try it next semester with a business calculus class. I’m also teaching abstract algebra and I have more trouble wrapping my mind around doing it in that context. On the one hand, I think that bombarding the algebra class with definitions might be an advantage in their working hard to learn them. On the other hand, well, I don’t have an “other hand” here; I don’t see a reason it shouldn’t work, too. These are dangerous thoughts you’re putting in my head!

    Since you weren’t offended at the previous editing of your Math 119 syllabus…

    page 5, line 12— “The each quiz…” should be “Each quiz…”
    Not sure if this is a typo. On page 6 line 13, you say, “Your audience is the set of 348 students, not me.” Are you referring to Math 348 (what I suspect) or do you have that many students in class (hopefully not!)? Since this is the Math 119 syllabus, the former would be confusing. I would write a LaTeX macro for the class number so that you can always match the number in that sentence with the title of the course on page 1 with having to change the number in both places each semester.


  4. Grading Epiphany (or, Bret was completely wrong about everything) « Solvable by Radicals Says:

    […] calculus last semester (and other courses in previous semesters), I had a general format of: “if you can do the basic skills from the course, you will get a C. To get a grade higher […]

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