assessing standards

As promised, today I want to describe my plan for assessing the standards in my multivariable calculus class. I’ve pretty much settled on the “common standards” that I think would be appropriate for any intermediate college math class, and thanks to some feedback I’ve received since yesterday, I’m refining the list of “content-specific standards” for this class. (For some of the reasons I’m using standards-based grading in this class, see this post, or these slides by T. J. Hitchman from last week’s Joint Math Meetings.) As I see it, there are 4 issues to deal with in scoring standards:

• what scale to use;
• how to assess;
• how to re-assess;
• how to convert to a letter grade at the end of the semester.

I’m almost scared to bring up the last one, because it’s the issue that could unravel the whole process, but I’m certain my (highly driven and motivated students) will panic without it being addressed. If there are suggestions for other issues that should be ranked with these, please let me know. I’ll cover each of these briefly.

What scale I will use

I’ve seen several proposals, including the very simplest, a 2-point system for each standard. (To be fair, I think that works when the list of standards is more refined, so that very specific skills are treated separately and not clustered.) After thinking about what I believe will be the most useful to students, and based on my experience using a 3-point system, I’ve decided to score each standard out of a possible range of 0–4, with 0 indicating “complete unfamiliarity” and 4 indicating “complete mastery”. To aid the students in seeing what I expect at each level, I’ve written sentences they should be able to read and agree with when assessed at the various levels. This is another idea that I’ve borrowed from somewhere, but am having trouble finding at the moment. In my syllabus, I’m describing a standard as a set of closely related skills that represent a piece of knowledge towards mastering the class material, which should explain some of the language below.

1. “I have some idea of what this skill set and its vocabulary mean, but I don't really know how to use it.”
2. “I can complete basic exercises that involve these skills as long as I have some guidance.”
3. “I can use these skills in familiar situations with generally good accuracy.”
4. “I can use this skill set effectively and explain its significance. I can recognize when the skills are useful and apply them to both familiar and new situations.”

(I did not write a sentence for 0-level, as it would be hard for someone completely unfamiliar with a topic to muse on her understanding of it.)

How I will assess

In brief, there will be homework, two midterm exams, and a final exam. All of these will be assessed on the basis of individual standards, and each time a standard appears, its new score replaces the previous score.

I know the debate rages on about whether or not to grade homework, but because the learning time is compressed in a college class, and I do not get to see my students everyday, I think it’s important to have some way to encourage and recognize work done outside of class. That said, the homework grades will not be based on “completion”. Instead, they will provide an opportunity for students to set a “base-level” for their understanding. The report from each homework assignment will list the relevant standards and how the student’s work rates on those standards. This gives them immediate feedback, as well as a chance to see how prepared they are in advance of the exams. I suppose a student could just copy someone else’s work to inflate their scores, but I will explain that in that case their Presentation score (which is part of every assignment) will suffer; their work should be original.

Exams are larger collections of standards, integrated into a broader context. By the time a student gets to a test, she should have a good sense of which areas she will do well in, thanks to homework and earlier self-assessments. Part of the review for each test will include a list of the standards that have been covered to date and may be expected to appear. (This is another good reason for my standards to be a bit coarse, rather than drilling down to specific types of computations—it’s easier to guarantee that a test covers “parametrized curves” than “parametrizing lines”, “parametrizing circles”, “parametrizing spirals”, “checking for smooth points of a curve”, etc.) Again, I suppose a student could not have done any homework before the test and demonstrate total mastery of the material, but that outcome is not, in principle, outside of my goals for SBG.

How I will re-assess

This will be tricky to explain. For many students, tests have always been about how much they contribute to the final grade, rather than how much they say about the current level of understanding. I want to make clear that tests are important and useful only insofar as they create a rich opportunity for learning (through synthesizing the material) and showcasing one’s abilities. Whereas homework assessment is intended to establish a base level of understanding about a student’s ability from week to week, an exam provides a snapshot of her ability, and often a stressful one, at that. After the test, I want to give every student a chance to prove herself in the areas where she may have previously struggled. The experience of other teachers using SBG suggests that this not be done indiscriminately.

Thus, my policy (initially) will be to have students contact me to schedule reassessments for specific standards (during or outside of my usual office hours), at any point in the semester after a standard has been tested. This reassessment could take the form of either an oral examination or an expository presentation by the student. It is unlikely that another written assessment will be given, since I believe the obstacle is often precisely that written tests provoke anxiety. No standard can be reassessed more than once a week, and no more than three two standards can be reassessed in a week. The main point among these practical considerations is that if a student proves she has mastered a course standard, then she receives credit for doing so.

How I will convert to a final grade

This is the least important of the four issues, and yet it is the one that leaves the most lasting record. (In contrast, I hope that what leaves the most lasting overall effect is the knowledge and confidence the students gain.) I don’t want to encourage students to fiddle with a fixed formula, especially since this is my first time using SBG, but I do want to make it clear that mastery of standards is directly correlated with the final letter grade. So here’s what I’m starting with:

• In order to guarantee an A in the class, a student should attain 4s on at least 80% of the course standards and have no scores below 3.
• In order to guarantee a B in the class, a student should attain 3s on at least 80% of the course standards and have no scores below 2.
• In order to guarantee a C in the class, a student should attain 2s on at least 80% of the course standards.

This emphasizes that the goal is mastery. It is also commensurate with what one might expect of the scoring levels in any case: “mostly 4s” should look like an A/A-, “mostly 3s” should look like some form of B, etc.

The score that will be counted for each standard towards the final grade will be the average of the latest score and the highest score. That way earlier gains will not be wiped out by later retreats, but it is still important to keep up each set of skills. Because there are no opportunities for reassessment after the final exam, any prior standards that reappear on the final can only be raised by the scores on that test, not lowered.

And that’s it! That’s my plan for assessing the 20–25 standards that will finally form the basis for grading multivariable calculus this spring. Thoughts and advice are welcome.