a reflection on course structure, and standards for calculus

Here’s what I’ve learned about writing standards: it’s hard to get them balanced properly. This challenge is inherent in developing any grading system. I used to fret about whether quizzes should count for 15% or 20% of the final grade; now I fret about whether the product, quotient, and chain rules should be assessed together or separately. (I’m happier trying to solve the latter.)

Another challenge is in setting up standards so that assessments have some coherence. I’ll explain. My first couple of times creating standards, I sat down and made a list of all the things I wanted my students to be able to do by the end of the semester, grouped into related sets, with an eye towards having each standard be of roughly equal importance (as I mentioned in the previous paragraph). After all, that’s what standards are, right? All the skills we want students to develop? That done, I told myself, “Okay, now every assessment—every homework, quiz, and test—will have to be graded on the basis of items in this list.” In principle, it’s nice to have this platonic vision of what students should do and know, including all the connections between related ideas (parametrization means imposing coordinates on an object; it doesn’t really matter what dimension it has, so parametrizing curves and surfaces should go together as a single standard). However, while this list said a lot about what I thought students should do, it didn’t say much about what I was going to do. It didn’t fit the structure of the course, just of the ideas (oh, wait, we’re parametrizing curves in week 2 and surfaces in week 10—why didn’t I notice that before?). Looking back, I can see that a lack of contiguousness within a standard does reflect a conceptual distinction between the concepts involved (hmmm, maybe the idea of drawing a curve through space is conceptually different from laying out a coordinate system on a curvy surface). I ended up assessing “partial” standards at various points in the semester, which is absurd on the face of it. It’s one thing to assert that a standard may be assessed at different points in the semester, based on how the skills are needed for the task at hand; it’s another to say, “Well, you’re learning part of a skill now, and I’ll test you on that, and you’ll learn the rest of this same skill later.”

I’ve had fewer slip-ups of this sort as time goes on, but I’ve never quite been happy with how the standards match up with the time spent in class. Both of the problems above keep rearing their heads. So for this fall, I decided to look at the schedule of the class and write standards based on what we do in 1–2 days of class. (Reading this blog post by Andy Rundquist earlier in the summer helped push me in this direction.) If it seemed like too little or too much was getting done in a day, well, that’s an indication that the schedule should be modified. In a semester with 38 class meetings, there should be sufficient time allotted for review, flexibility, and a few in-depth investigations, which leads me to having 25–30 content standards for the course. That’s a few more than I’ve had in the past, but not by many.

Here’s the conclusion I’m coming to: standards both shape and are shaped by the structure of the class. Part of what we as instructors bring to a class is a personal view of how the subject is organized and holds together. If you and I are both teaching calculus, there will be a great deal of overlap in what skills we believe should be assessed, but there will be differences, and we’ll find different dependencies. A fringe benefit of writing out standards is that we can see this structure clearly—even better, I believe, than just by looking at the order of topics. They force us to be honest about our expectations, thereby combatting a certain tendency, observed by Steven Krantz in How to Teach Mathematics, to give tests based on “questions that would amuse a mathematician—by which I mean questions about material that is secondary or tertiary. … In the students’ eyes, such a test is not about the main ideas in the course.” You may want students to use calculus mostly in applied settings where exact formulas for the functions involved are not known, whereas I may be primarily concerned with students’ ability to deal formally with closed-form expressions and to deeply understand classical functions. We can both be right. We should both let our students know what we expect of them, rather than making them guess. In short, standards are not completely standardized—they highlight the commonalities and the particularities among courses that treat basically the same material.

With all that said, here I will share my list of standards for Calculus 1 this semester. Because of the length of the list, I’ll just link to a Google document that contains them: Standards for MTH 111, Fall 2014. They are grouped into twenty-six “Content standards” and three “General standards”. Over time, I’ve settled on these last three as skills that I want to assess on every graded assignment: Presentation, Arithmetic and algebra, and Mathematical literacy and numeracy. These are essential skills for doing anything in calculus, and struggles in calculus can often be attributed to weaknesses in these areas. We’ve all had students who are fine at applying the quotient rule to a rational function, but are stymied when it comes to expanding and simplifying the numerator of the result. That can hamper solving certain kinds of problems, and I want to be able to point to “algebra”, not anything calculus-related, as the area that needs attention. The descriptions of the content standards are shaped in part by our textbook, Calculus: Single Variable by Deborah Hughes-Hallett et al. I like to introduce differential equations fairly early in the course—this follows a tradition at my college, too—so some standards related to that are sprinkled throughout. I should also confess an indebtedness to Theron Hitchman for the language of using verb clauses to complete the sentence “Student will be able to …”

In addition to the 29 standards in the document linked above, I have one more for this class: Homework. Oh, homework. The calls to treat homework purely formatively and to stop grading it (link goes to Shawn Cornally’s blog) have not quite reached the halls of post-secondary education. Many college and university instructors believe homework is so important that they make it worth a substantial fraction of the students’ grades. And it is important, but solely as a means for practicing, taking risks, developing understanding, and making mistakes. (See this video by Jo Boaler* on the importance of making mistakes: “Mistakes & Persistence”.) Grading homework almost always means that its usefulness as a place to take risks is undermined. Last semester I didn’t grade homework at all, although I did have a grader, who made comments on the homework that was submitted. On average, about 1/3 of the class turned anything in. At the end of the semester, I got two kinds of feedback on homework. A few students expressed appreciation that the pressure to make sure that everything in the homework was exactly right was relieved. Several, however, said they realized how important doing homework is to their understanding—often because they let it slip at some point—and urged me to again make it “required”. I want to honor both of these sentiments. I want to encourage students to do the homework and to feel like it is the safest of places to practice and make mistakes, and thereby improvements. So I will count both submissions and resubmissions of homework towards this standard. A student who turns in 20 homework assignments or thoughtfully revised assignments will earn a 4 on this standard, 15 will earn a 3, and so on. I hope this will have the desired effect of giving students maximum flexibility and responsibility in their own learning, while also acknowledging the work and practice they do.

All of the rest of the standards, general and content, will also be graded out of 4 points, with the following interpretations: 1 – novice ability, 2 – basic ability, 3 – proficiency, 4 – mastery. (I’ve adapted this language from that used by several other SBG instructors). At the end of the semester, to guarantee an “A” in the class, a student must have reached “mastery” in at least 90% of the standards (that is, have 4s in 27 out of 30 standards), and have no grades below “proficiency”. To guarantee a “B”, she must have reached “proficiency” in at least 90% of the standards, and “basic ability” in the rest. A final grade of at least “C” is guaranteed by reaching “basic ability” in at least 90% of the standards.

Two other blog posts about standards in college-level math classes went up yesterday:

• Bret Benesh wrote about his near-final list of standards for calculus 1, and again explained his idea to have students identify for which standards they have demonstrated aptitude when they complete a test or quiz. I really like this idea, as it essentially builds metacognition into the assessment system. I will have to consider this for future semesters.
• Kate Owens posted her list of standards for calculus 2, which she has organized around a set of “Big Questions” that highlight the main themes of the course. This is particularly important in calculus 2, which can sometimes seem like a collection of disconnected topics. In an ensuing discussion on Twitter, it was pointed out that these kinds of Big Ideas are what can really stick with students, far beyond the details of what was covered.

After reading Kate’s post, I looked at my monolithic list of standards, and attempted to organize them into groups based on three big questions: “What does it mean to study change?” (concepts of calculus), “What are some methods for calculating change?” (computational tools), and “What are some situations in which it’s useful to measure change?” (applications). I was not particularly successful at sorting my standards into these categories, but I like the questions. I may ask the students how they would use the various standards to answer these questions. There are trade-offs in any method of developing a set of standards. I am grateful for these other instructors who are also working on changing how we think about grading and sharing their ideas.

* Jo Boaler’s online courses on “How to Learn Math” are currently open:
For teachers and parents until October 15 ($125)
For students until December 15 (free)