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I am a mathematics professor seeking advice on refining my grading system for future courses. I currently employ a bonus points system intended to provide a catch-up mechanism for students who may be falling behind. However, I've noticed a trend where the bonus points disproportionately benefit students already doing well, leading to a grade distribution heavily tilted towards the higher end.

I do not wish to curve grades due to the competitive environment it might foster. What I'm seeking are strategies to make the attainment of an 'A' grade more challenging while still providing support for struggling students.

Here are a few ideas I've considered:

  1. Adjusting the bonus points system so that students at the lower end of the grade spectrum receive more benefit.
  2. Offering remedial assignments for students who are falling behind.

I would greatly appreciate insights or experiences from other educators. Are there additional strategies that have worked well in your own classrooms?

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    $\begingroup$ bonus points disproportionately benefit students already doing well -- This has been my experience also, probably more so because I tended to give "bonus points" more for tangential topics or more advanced work than for extra work on basics (and I called it "extra credit"). That said, maybe some of the things I used to do could work for you -- see this answer. $\endgroup$ May 17, 2023 at 14:31
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    $\begingroup$ One challenge students who have fallen behind face is not feeling like they can catch up/not having clarity about how to catch up. In general, you'll want to pay attention to what the paths are for students who are struggling to catch up, how feasible those paths are, and whether students are actually properly aware of them. Is your bonus points system actually practical for catching up? Are struggling students aware that they should be using the bonus points system? $\endgroup$
    – TomKern
    May 17, 2023 at 23:45

5 Answers 5

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Probably the most straightforward mechanism is to potentially replace some of the grades in the gradebook, e.g.:

  1. One or more assessments (tests, quizzes, assignments) are dropped from the averaging formula, counting only the higher ones.
  2. Assessments may be re-taken, with only the higher grade used in the final average.

Both of these methods allow low-scorers to boost their grades, while having little or no benefit to those who scored high in the first place. They also serve to handle missed assessments due to illness or other interruptions. And both of these mechanisms are implemented in standard online learning management system gradebooks.

Note that #1 makes no extra labor on the part of the instructor, while #2 does so.

(Personally I do #1 but not #2.)

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    $\begingroup$ (+1) for "Note that #1 makes no extra labor on the part of the instructor, while #2 does so." I suspect new teachers don't sufficiently take into consideration how much TIME various teaching methods involve when making decisions about using them. $\endgroup$ May 17, 2023 at 14:46
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    $\begingroup$ I also do #1 and almost never #2. The extra advantage is that I can announce the "no make-ups" policy and the students who have 1 or 2 bad days or family emergencies do not need to retake anything and I do not need to check which absence excuses are legitimate and which aren't. Everybody is welcome to be at their worst on a few quizzes or to skip them altogether. $\endgroup$
    – fedja
    May 17, 2023 at 22:51
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    $\begingroup$ #1 is problematic with regard to assessing learning goals and supporting struggling students. $\endgroup$
    – user1815
    May 18, 2023 at 6:14
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    $\begingroup$ @quarague To me, you describe an idealized situation, more accurate of some small primary school classes I've witnessed (although they were mastery-based) than large college classes. By "problematic", I did not mean just plain wrong; rather: nuanced, or more precisely, "creates a problem to be solved" (and usually it can be solved). Also by nuanced, I mean too complicated to discuss in comments (just the variation in class types, e.g.). One criterion above is "no extra work," and the implied "support" seems to be numerical grade relief, not improved learning. That needs to be unpacked. (1/2) $\endgroup$
    – user1815
    May 25, 2023 at 15:39
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    $\begingroup$ You'd also have to unpack how knowing where support is needed translates to effective support. I drop assignments, but I design overall assessment to support the learning goals (that's the problem to be solved in "problematic"). One tenet is that in college, in my classes, "learning is important." It's not a game in which I have to create backdoors to get around some horrible monster called minimal competency. I've also seen dropping tests work to the detriment of the students, good & struggling, and that makes me sad. (2/2) $\endgroup$
    – user1815
    May 25, 2023 at 15:40
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If you'd like some reading on this, I suggest Feldman's "Grading for Equity", which goes through the principles of trying to get grades to reflect students' final demonstrated understanding of the course goals. This means if a student is unable to demonstrate a learning objective on an assessment, they should have plenty of opportunities later in the semester to demonstrate the objective, and thus to make up the full points. This encourages students who have fallen behind to focus on learning the topics they've missed. In fact, a students' best strategy in this system is often to focus on the most fundamental learning objectives, and with a little guidance you can help students see that.

I implement a makeup quiz system: grades are 100% quizzes. Each quiz question covers a particular learning objective, and students have many opportunities to get the point for that question throughout the semester.

Some things I've found to make makeup quizzes less time consuming:

  • Grade each question 0 or 1: either the student demonstrated the tested skill or not. Full credit for unimportant mistakes.
  • Makeups should be on an individual-question basis. Only give students questions they didn't get full marks on already. Canvas LMS has rubrics you can use to keep track of which problems students have the point for, and when you update the rubric scores it automatically updates the quiz score.
  • Keep each question as simple as possible to test the desired skill. This makes it easy to come up with new versions of the question.
  • Unless students have trouble understanding the original question, just rerandomize the numbers/formulas, don't bother coming up with a completely different question for the desired skill. This isn't a great compromise, but it's necessary to keep workload reasonable.
  • Eventually you'll want lots of students to be taking lots of makeups. It's a great way to encourage students to keep studying a topic until they get it. Try to get lots of students working on makeup quizzes simultaneously. Once you've sped up the problem creation and grading processes, proctoring becomes the biggest timesink. Try to get lots of students in office hours at the same time or offer time in-class for makeup quizzes.
  • One way to reduce end-of-semester demand for makeups is with "makeup midterms/finals". Create a makeup version of every problem from previous quizzes, then have students just make up the problems they need to make up in the style of an in-class exam.
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  • $\begingroup$ This seems to work well for numeric problems. But for proofs it's not really possible to make similar problems, right? $\endgroup$
    – faceclean
    May 21, 2023 at 1:21
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    $\begingroup$ My advice is for problem-based math courses, but the general principle applies to proof-based. Focus on your course goals. If you're trying to test a particular proof, all you can do is re-ask the problem, which is what you want to do if having done that particular proof is a desired outcome (i.e. it's an important proof). On the other hand, if you want to test specific proof skills, usually those skills will be tested later on in the course anyway. You can have a rubric for the whole course keeping track of which specific proof skills students have demonstrated to what degree. $\endgroup$
    – TomKern
    May 21, 2023 at 1:49
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If you truly are interested in putting the time and effort into helping low-to-middle students learn the material and achieve success, then I highly recommend Standards-Based or Mastery Grading. Here's a link to a document that inspired and guided me in the beginning: "Building Calculus: The grading system". There are abundance of materials to find through web searches once you know the right phrases.

I have used a standards-based approach in my lower-level courses: First semester STEM calculus, business calculus, finite math, and (non-proof) discrete math. The results are profound. A-level students don't mind the weirdness. And any other grade-level student can always see a pathway to improving their grade if they so desire.

There are downsides. One reason I don't use it in my upper-level classes is you have to have an abundance of assessments/problems. (This can be hard to generate in proof-based courses.) Office hours will be busier (especially as the semester's end approaches). And the gradebook system will be complicated (I always enjoy a good spreadsheet challenge).

But in the end it's all worth it. Students who want to put in the effort to learn the material and improve their grades will do so. (The students who don't want to put in the effort... well, the outcome is the same.)

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It has also been my experience that most bonus schemes disproportionately benefit stronger students and increase grade inequality. I think that dropped scores (as mentioned in Daniel's excellent answer) are a helpful start, and I use them in all of my courses.

But going further, I find it helpful to think about why I'd like to give bonus points. As you mentioned, I would like bonus points to serve as a "catch-up" mechanism for weaker students. I don't want to inflate their grades artificially, but I want to help them make progress and also feel like they're making progress.

This lead me to the following:

  1. I aim to use bonus points to "subsidize" behaviors that promote engagement with the course objectives.
  2. I usually tie available bonus points to the number of lost points on an assessment.

Some of the answers above talked about make-up assessments, but I find managing the logistics of these frustrating and tedious. Instead, I've done the following:

  1. Give students that opportunity to complete an exam reflection sheet with bonus credit being awarded based on original test score (90-100% gets +1%, 80-90% gets +2%, etc.) Here's an example and more detailed description: link.
  2. Give students the opportunity to recover some percentage of lost test points by attending tutoring sessions at the university STEM Center. For example, I allow students in math for business to recover up to 30% of their lost points. This keeps the strong students from wasting their time, but it can make a huge difference for students who are at the bottom of the class and need to spend more time intentionally engaging with the homework. (I don't do this for Calc 1, as it could allow ill-equipped students to continue to Calc 2.)
  3. Allow students to recover full credit on up to $n$ exam problems from an assessment by presenting the solution(s) to me on the whiteboard in my office without any notes. (This is a big time commitment for me, so I've only offered it in upper-level classes when I've given an exam with a poor class average and only for one problem of each student's choice.)
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My idea is that you should just give the students a second chance on each problem/task without adding any workload for yourself. The extreme scheme would be like this: you give, say, 10 quizzes of 3 problems each, but with only 15 problems total (just literally, without changing even the numbers, but, of course, with a noticeable time spacing between the identical problems and without a warning about what exactly will be repeated). Then you just use the best attempt grade on each of those 15 problems. That extreme is a bit hard to implement exactly as it is because one needs to more or less align the quiz problems with the material being covered, but you can modify it a bit in various ways so that it becomes workable. The students who solve the problem from the first attempt for the full grade do not even need to bother to try it again and may spend more time on the new ones, which is a good incentive to do things right from the beginning for those who have some common sense and elementary logic. You can also include homework or quiz problems into exams and if they are solved correctly on the exam, upgrade the initial scores.

Another idea that works in a small (10-20) people class is to give them a number of problems in the very beginning that they can submit any time during the course making as many attempts as they wish (if you do not want to endure too many attempts, you can set up a waiting period after each attempt or say that each problem can be tried at most once within the same week). When I was a student and we studied integration, we were just given a list of 100 integrals from the start some of which we didn't learn to do until later (unless one chose to study the corresponding techniques by him/herself ahead of the schedule). The submission was oral. The requirement was that by the time of the submission, the student should be able to do it without outside help (the exact way to ensure that varied but a common technique was just to write the integral on a blank sheet of paper and to let the student do the full computation in front of you). No real attempt to restrict what help the student was getting outside the classroom was made (everybody agreed that there was no point to try to control what one could not control and that the surest way to make a person dishonest is to make him/her promise absolute honesty). I use this scheme in my graduate classes all the time and in undergraduate ones quite often, though the portion of the total grade coming from such assignments is lower in the latter case.

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