Conjunctive Standards-Based Grading

I’ve had a lot of requests lately to explain my grading system, so I thought I would outline it in as much detail as possible here. I learned this summer that what I do is apparently called “conjunctive scoring”. That is, students cannot compensate for a low score in one area (say, their understanding of forces) with a high score in another area (say, conservation of energy). Rather, it requires at least a minimum amount of mastery in every area to earn a passing grade overall.

A minimum amount of mastery?

My physics objectives are split into “A” and “B” flavors. At minimum, each student has to demonstrate consistent mastery on the A level objectives. So grades between 70 and 90 depend on how many of the B level objectives a student has mastered. More on how I get the number grade below. The important part for now is that it is clear to students whether each skill is an A or a B skill from the start, and that A skills are the required, more basic (though not necessarily easier) objectives.

Even if a student has shown mastery on some B’s, if they are missing an A objective, they are not yet in a position to earn a passing grade at the end of the term. Luckily, their grade doesn’t exist until the end of the term, so they have plenty of time to do corrections and pick up skills that they missed along the way. During the final week or two of the quarter, I remind students which A objectives they have yet to demonstrate, and I start making sure (writing comments to advisors, etc) that they come in for help and then to assess.

Quick note on my naming convention: I didn’t want the name (A, B, etc) to conote a letter grade, I wanted to make it easy to have as many levels as needed (turned out 2 is right for now, though I started out trying three last year), and there are too many numbers involved already. Of course other naming conventions would be possible (colors? people? objects? etc).

How I grade assessments: Yes/No and Feedback

A scanned copy of a grading sheet from last year. This is a typical example of what the front page of a returned test looked like. Click to see it bigger.

First, I work through every test, marking them up with comments, corrects, etc. Since I don’t have to worry about points or grades, this process goes pretty quickly for me, even when I’m writing feedback on their work as I go.

I use a binary system to score each objective for each student on each question (where it is relevant). In the picture, I have a scanned copy of the scoring sheet I attach when handing back a test. In the rows going down are objectives, in the columns going across are question numbers. The final column shows the overall recorded score for each objective. I only record one score per objective per assessment, even though I usually measure each skill multiple times on the test.

The markings that I use:

2 = Mastery shown (this is a “yes”)

1 = Developing mastery— could be an error in process, arithmetic, units, etc, but something about the approach was correct. (this is a “no”)

0 = No mastery shown— so many errors or confusions that the student does not seem at all close to mastering this skill. (this is a “no”)

– = No data— student misinterpreted a question so much that the skill I’m trying to test is not observable in their response, or I don’t see their response as good evidence either way, or their response simply did not involve the skill. It is sometimes possible to have a completely correct solution without showing a particular skill that I was expecting to see.

In the final column, I put their overall score. If I recorded a 1 for a particular skill on any question, then the overall score is a 1. Even if the majority of scores are 2’s, I still record it as a 1 because I am looking for any evidence that a student has not mastered the skill. If I were to “do them a favor” and ignore the 1 in favor of two or three 2’s, then I would actually be setting them up to fail down the line because I’m letting them ignore a problem (even if it is, or they think it is, a small one) that they will likely continue to have.

As shown in the scanned example, sometimes a student shows mastery (or in this case, inadequate mastery) on an objective that I didn’t anticipate. It is easy to simply write those in at the bottom of the sheet.

At the end of the marking period, I only look at the most recent score when determining a number grade. (This process gets a bit more nuanced when the end of the marking period includes an exam, but more on that later.) Scores can go up and down as more data is accrued, and the scores do tend to fluctuate (which I take to mean that one data point is not sufficient for measuring complete mastery on many skills). I think letting scores go back down is an especially important piece of this grading scheme, and I’ll talk more about that below.

Important note about a difference between A and B objectives: To get a 2 for an A objective, the student must show that skill perfectly in the problem. To get a 2 for a B objective, the student typically must both show that skill perfectly and get the problem completely correct. That is, you can get problems wrong but still get “credit” for the A objectives, but to get the B objectives, you must be able to do the entire problem consistently. So you will not be stopped from passing the class if you never learn to use your calculator in a proper and repeatable way, but you will not be able to get an A in physics if you can’t finish all of the problems correctly. This idea both seems reasonable to me, and also seems (after one year of trying it) to result in much more careful students who are much better at doing arithmetic and routine calculations.

What if I get a 1 or a 0? How additional assessments work.

On the first test for a topic, more than a couple of students get all or many 1’s (while a 0 is not uncommon on one problem, often students still get a 1 overall because they have shown some developing mastery on another problem).

Now that they’ve gotten some feedback, it is time for students to start remediating and improving. This work comprises the majority of the out-of-class work, aka homework, that I’m giving them and is very self-directed. Still, I help them develop the tools and process to address their mistakes so they are not completely on their own.

The first step is to make corrections to their work. Often they will check in with me briefly once they have done that (usually at breakfast, a nice benefit of boarding school) to make sure what they have done is now correct. Next, they need to additional practice. The next assessment on the same skill will probably look very different to them, so they need to make sure that they learned the skill itself, not just how to answer the original question. This year, I am putting extra practice problems (plus answers) on a class website.

Then, when they are ready, I will have them apply for an additional, out-of-class assessment. I’ve blatantly stolen Sam’s application and modified it for my needs here. Thanks, Sam!

This year, I am only giving these opportunities once per week. I am also planning to give in-class assessments once per week instead of waiting until the end of each unit. I will vary the length (sometimes 10 minutes, sometimes full period tests). Everything that we’ve done so far in class will always be fair game. I hope this change will improve the data that I’m taking and help make students feel more comfortable with assessment as a part of the improvement process.

What does it mean to earn an “A”?

In my system, an end grade of “A” represents mastery of all the objectives detailed for each unit. To get beyond a 90, though, they need to move past the atomized skills and show synthesis. They must know when to use each model, must be able to use multiple models for different parts of one motion or problem, and must show creativity in their thinking.

I try to measure this depth of understanding on the semester exams in two ways: by seeing how they approach a series of comprehensive but traditional physics problems and by using goal-less problems. Before the exam, they might have shown mastery on skills in a somewhat isolated way. They apply for an additional assessment on specific objectives, then I give them questions that address those requests. On the exam, they must demonstrate most or all of the skills on problems that are not categorized for them as belonging to a particular model.

I wrote about my semester exam last school year, and I plan to write an update after going through the same process one more time this coming January.

Calculating the semester grade

In the end, I have to turn all of that rich information I collected into one final number.

Here is the basic plan:

I’ll start by assuming all students have demonstrated all A objectives going into the exam. I make it pretty difficult for them not to do this by the time exams start, and I have the advantage of students living here when I am tracking down those final few students.

On the exam, I give them a print-out of all of the objectives from the semester. Any that they have not yet demonstrated, I highlight for them. They turn this paper back in with their test and I use it for grading.

If they demonstrate a highlighted skill, I cross it off. If they falter on one they had already shown, I circle it. For the circled skills, I look back at their history over the semester (thank you, ActiveGrade). I then have to decide whether their mistake on the exam outweighs a consistent history of mastery. More often, though, a mistake on the exam corresponds with an inconsistent history on that skill. Sometimes they perform a skill correctly on one problem and incorrectly on another during the exam. After looking at their history and all of their work, I decide whether to count that skill as a yes or a no (or sometimes if they can do it correctly on an easier problem but not on a harder one, then I’ll count it as a 1/2 yes).

In the end, I count up the number of missing B objectives (counting any 1/2 yeses as 1/2 of a B objective) and subtract it from the total number. I use that percentage of B objectives to interpolate a score between 70 and 90. So mastering half of the B objectives would correspond to a final grade of 80.

To get the final 70 to 100 score, I also take into account their work on the goal-less problems. And in the event that their performance on the exam is much worse than what they were showing me on smaller assessments (this is very rare), I also have a means to take into account the earlier data that I collected. I describe that in my earlier post, so I’ll end the calculating grade discussion here.

Why I think this grading system rocks

I’m sure I will miss some of the reasons, but here are a few highlights of why I love grading this way. I’ll also focus more on why I especially love this particular flavor of SBG. For more reasons why the whole idea of grading in a feedback loop of formative assessment and remediation based on standards is amazing, read Shawn’s blog.

• For the most basic skills in my class (about 9 out of 30 total in my regular class), it means there is no moving on. It’s not okay to never “get” a topic. It is a promise to my students that they are going to learn at least this much physics, and that I will keep working with them on it all year.
• It helps struggling students see where to focus their energy first (start with the A objectives). It rewards starting problems even when you are sure you won’t be able to finish them (and often students find that once they start, they actually CAN finish).
• It raises the expectations of learning for everyone in the class. If 100 in a class means that you’ve mastered all of the skills, then the expectation is that not everyone will do that. No school is going to be okay with everyone in the class getting 100 (or even with a many in the class getting 100). More on that in a minute. Students are essentially encouraged to be satisfied with (or even happy about) a very incomplete understanding of the content covered in the class. Moving that pin down to a 90 means that there is an expectation that many (I really hope ALL) students will master the skills that we learn in class. The 90 to 100 now represents doing something extra with those skills once you have them.
• It exposes cheating as the pointless, silly exercise that it is. There is little point to cheating on a test when your score on that test will not affect your final grade (since all of the future tests of the same skills will bury that score).
• It devalues “cramming” because while that may work for one test, it won’t keep working, and it definitely won’t work on the exam.
• Students feel strong and powerful with their skills. They have demonstrated them over and over. There is no way to explain their success away as luck. So even though the exam is still a bit intimidating, they are able to have a more substantial faith in their tried and true abilities.

Tangent time: can you think of a practical class that people would want to take where we would be unhappy to have a 100% average? Example: swimming class, CPR class, driver’s ed, cooking class, etc. So why, for a class like physics that many are perhaps not taking in a completely voluntary way, do we not want them to all learn everything? Why is our expectation that they not all be completely successful (in fact, our expectation is even worse: that no one will be completely successful)? Perhaps this is a post for another time, though.