This is a continuation from my previous post.

I have been unhappy with grades in general (I cannot believe that I cannot find a link to a previous post where I complain about grades). However, grades at my school are not going away any time soon, so I am working on improving them this semester.

Many thanks to Jason Buell, Shawn Cornally, Dan Meyer, Kate Nowak, and Mark Olson for helping me get my thoughts in order.

Here are my major complaints about how grades are typically done in math classes:

1. They are not informative. What does a “C+” or “73%” tell you about what you need to improve upon? It tells you roughly “how much you know,” but not “what you know.”
2. Grades are designed to be incentives, but they incentivize the wrong thing. A student who does poorly on a quiz is actually better off (prima facie, at least—and our students often never get past this stage) ignoring the material on that quiz to concentrate on learning the material that will be on the next quiz.
3. I cannot find the source, but one math department chair lamented that his students could get a major in mathematics without every getting a problem correct in his/her four years at college (e.g. you could get 7/10 on every problem and get a degree).

I suppose that one advantage that our current system has is that it allows us to easily rank students according to “ability.” I have doubts as to how effective this is, but—regardless—I do not see a good pedagogical reason to rank my students. It gives neither them nor I information about how much mathematics they know.

Here is my solution (with lots of help—see the bibliography above):

1. I listed the content of the course into “topics.”
2. Have frequent quizzes. This has the advantage of making assessment more familiar and less scary.
3. Each quiz question is graded and (essentially) recorded individually. There are only two possible grades: “Acceptable” and “Incomplete.”
4. If a student gets an “Incomplete,” he/she must either get a similar problem correct on a later quiz OR come to my office and have a make up quiz. There is no limit on the number of redos (save for the time limit of “end of the semester”). There is no penalty for missing a quiz question (except that you have to make it up later).
5. The questions are split into two types: “skills” (e.g. taking the derivative of $x^2+3$) and “concepts” (e.g. demonstrating understanding of where the definition of the derivative comes from).
6. The students need to get an “Acceptable” for each skill topic three times; once this has been done, he/she can skip all later questions concerning this skill topic.
7. Similarly with concept topics, but they only need to do it twice.
8. A student earns a C for the course if he/she gets “almost all” of the “skill” requirements met.
9. A student earns a B for the course if he/she gets “almost all” of the “skill” requirements met and roughly half of the “concept” requirements met.
10. A student earns an A for the course if he/she gets “almost all” of all questions requirements met, “almost all” of each topic requirements met (so he/she must have no weaknesses), and do a project.

There will be a quiz once per cycle (roughly a week), as well as “team quizzes.” These team quizzes are done by choosing one member at random (once they feel they are ready) to do the quiz for the entire team. If the chosen person gets it correct, all students in the team get credit. Otherwise, all students on the team will have to redo it at some point (which they would have to do anyway).

There are other details. For instance, students are only allowed to redo one problem outside of class each day (for my sanity), there will be a midterm and final averaged in (for political reasons), but this is the gist.

I have been working on this post for two weeks. I am going to post it now;I wish I could spend more time on it, but I need to get on with my life.

### 14 Responses to “New Grading System”

1. Benjamin Hogan Says:

We are talking a little about grading in Math Pedagogy . I too feel your frustration with grading methods. In every math class I took up to your MATH 180 I prepared for tests with the memorize and forget method. In fact, up to MATH 180 I was convinced that I stunk at math. Now I realize I am just bad at memorizing, but at problem solving I am decently skilled.

• bretbenesh Says:

Hi Ben,

It surprises me that were not very good at previous math classes. This is further evidence that we are not running our math classes right—I would guess that most people would agree that it is better that a citizen be a good problem solver than be able to memorize.

Get to work on that when you graduate, will you?
Bret

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5. Joss Ives Says:

Hi Bret,

As promised, I have some questions for you…

Team quizzes: Can you explain this a bit more? Do they all prepare together as a group and then one person is chosen randomly to execute what they discussed?

Midterm and final exam: You mention that these are averaged in. Do you take the A/B/C from the skill-topic work and then average that with the exam scores?

The fact that you give them a midterm and final allows you to sidestep this issue a bit, but do you have any standards where part of the assessment is for them to recognize what skill/concept they should be using/applying? The kind of thing where they are given a problem and have to figure out which concepts or techniques from the course apply?

6. bretbenesh Says:

Hi Joss,

It is nice to hear from you.

Team Quizzes: students are in semi-permanent groups of three (sometimes two). If everyone in the team understands the material for the day, they can opt to take a “team quiz.” When they request one, I randomly choose one of the team members (with a die), and they take the quiz for the entire team—they all get the same outcome.

The problem I had is that there was no penalty for taking the team quiz. This initially seemed like a good thing, but the students took advantage of it. If I do it again, I might have they wager a small amount: they get two credit/points/whatever if the team succeeds on the team quiz, and they lose one if they do not. This might help teams focus more on learning.

But you basically got it right: they prepare together, and one person is chosen randomly to execute what they discussed. Rather than just typing “yes, you are correct,” I decided to give you your money’s worth with two extra paragraphs.

Exams: I average them with the skill-topic work (which I called “quizzes”). I did this by “GPA” and (student-determined) weights. So I converted all exams and “quizzes” to a 4.0 scale, and then did a weighted average. I am open to suggestions on how to improve it; I do not love it.

For your final paragraph: actually, the midterm and exam did not allow me to sidestep the issue, since they doubled as “quizzes” (and hence had the appropriate topic written next to the problem). This bothered me from the start, although it turned out to be not too bad (in an “eigenvalues” question, they still need to know to apply the determinant). My only idea to remedy this—and I do believe that it is a problem—is to give them larger projects where they need to solve the problem AND determine which ideas they used.

Do you have any help in solving the problem of synthesis on quizzes/exams?

That being said—I did give them some ungraded projects at the end of the semester that did not tell them which technique to use, but I did not have them explicitly determine which topics were used in solving the problem. It would have been better if I had.
Bret

7. bretbenesh Says:

Hi Joss,

Essentially, you are asking about how to incorporate “problems’ rather than “exercises.” I will use this language below.

I made “problems” a separate “topic/category/standard.” Of course, I called them “proofs,” as I am a mathematics teacher. But I think that the idea is the same. So really, I had four different types of assessments in my course:

Exams (the boring, usual exams; nothing SBG about them)
Quizzes (this is the SBG part of the course)
Proofs (this plays the role of your “problems”)
Project (they had to do one, but it is not relevant to their discussion)

So I had the students do their synthesis work on the “proofs” portion. Here are the logistics: I assigned one per week. The students submitted them, and I gave them feedback. There were two grades: “Acceptable,” and “Incomplete.” An “Incomplete” means that they can resubmit again for no penalty. Repeat until they get an “Acceptable.”

Here is how they affected their final grades: my general grading practice has been “the more evidence you provide me, the higher the grade.” In very simplified terms, this means that if a student demonstrated that they could do a standard five times, they got an A. Four times got them a B, and three times got them a C (this is VERY oversimplified).

The “Proofs” worked similarly. If they successfully solved all of the proofs, they got an A for that part. If they are only missing one, they get a B, etc.

In the end, I made a judgement call for the final grade. If everything was at an A level except for one B, then they got an A. There was no rigid formula (I was clear about this in the syllabus).

So that was my solution. I am interested to hear what other people came up with.
Bret

8. Joss Ives Says:

Brett and I are cross-talking across this post and one on my blog

It’s funny. I started to reply to this comment thread, but bigger ideas started popping into my head which lead to the post about Physics problems and SBG. But now I’m back to finish the reply that I started.

Hi Joss,

It is nice to hear from you.

Team Quizzes: I was thinking that this might be something that they could take advantage of. A student could feasably earn multiple “Acceptables” on a standard for which they were never assessed in this system. I love the idea of getting students to wager on things, but I am sure that there are some students that would hate it. On a side note, I do group quizzes in my classes where they all work together and the group portion counts for 25% of the quiz grade where the individual portion (completed first) counts for 75% of the quiz grade. They’re a great learning tool and I’m still trying to figure out how they could be brought into an SBG implementation.

For your final paragraph: actually, the midterm and exam did not allow me to sidestep the issue, since they doubled as “quizzes” (and hence had the appropriate topic written next to the problem). This bothered me from the start, although it turned out to be not too bad (in an “eigenvalues” question, they still need to know to apply the determinant). My only idea to remedy this—and I do believe that it is a problem—is to give them larger projects where they need to solve the problem AND determine which ideas they used.

Do you have any help in solving the problem of synthesis on quizzes/exams?

That being said—I did give them some ungraded projects at the end of the semester that did not tell them which technique to use, but I did not have them explicitly determine which topics were used in solving the problem. It would have been better if I had.

I like the idea of getting them to take responsibility for identifying some of the standards/topics being assessed. Andy Rundquist did something along these lines in his collaborative oral assessments by having the rest of the class discuss which additional or other standards (beyond the initial one) were addressed by a student when answering one of his oral questions.

9. Joss Ives Says:

Brett and I are cross-talking across this post and one on my blog

It’s funny. I started to reply to this comment thread, but bigger ideas started popping into my head which lead to the post about Physics problems and Standards-Based Grading. But now I’m back to finish the reply that I started.

Team Quizzes: I was thinking that this might be something that they could take advantage of. A student could feasably earn multiple “Acceptables” on a standard for which they were never assessed in this system. I love the idea of getting students to wager on things, but I am sure that there are some students that would hate it. On a side note, I do group quizzes in my classes where they all work together and the group portion counts for 25% of the quiz grade where the individual portion (completed first) counts for 75% of the quiz grade. They’re a great learning tool and I’m still trying to figure out how they could be brought into an SBG implementation.

“Do you have any help in solving the problem of synthesis on quizzes/exams?” Honestly, issues related to synthesis are one of my largest problems in trying to develop a picture for my own Standards-Based Grading implementation. Even in my current system I don’t think I do the greatest job of helping them work on synthesis, but I do feel like I have done a decent job of making it so that they have to demonstrate it on exams through multi-concept problems and explain-your-reasoning questions.

I like the idea of getting them to take responsibility for identifying some of the standards/topics being assessed. Andy Rundquist did something along these lines in his collaborative oral assessments by having the rest of the class discuss which additional or other standards (beyond the initial one) were addressed by a student when answering one of his oral questions.

10. bretbenesh Says:

Hi Joss,

First, thanks for clarifying the cross-talking—I should have provided a link myself, but I am lazy.

Second, Andy’s weblog has already convinced me to do collaborative oral assessments of sorts. You physics people have good ideas, and I enjoy stealing them from you.
Bret

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