By Priscilla Bremser, Contributing Editor, Middlebury College
By the end of every workshop and conference session on Inquiry-Based Learning that I’ve attended, someone has raised a hand to ask about coverage. “Don’t you have to sacrifice coverage if you teach this way?” Of course coverage took center stage for many of my professional conversations long before I tested the IBL waters; it’s important. But an equally important question is this: What do we sacrifice when coverage dominates? It may well be conceptual understanding; it’s possible to cover more ground, albeit thinly, if we settle for procedural understanding instead. More than once I’ve settled for even less, delivering a quick lecture just so that my students will have “seen” a particular idea. How do we strike a balance between coverage and other considerations when we are so practiced at reducing a course description to a list of topics?
Strong arguments for striking that balance have been made elsewhere. For example, Stan Yoshinobu and Matthew Jones offer a close examination of the “price of coverage”. “Coverage versus depth” is a “false dichotomy,” they say; racing through material makes for a passive student experience, which affects student understanding of what it is to learn mathematics. “Implied messages are sent to students through classroom experiences,” and some of those messages may have unproductive consequences, including overreliance on mimicking the instructor and memorization, and significant difficulties with non-routine problems.
Is there, on the other hand, a price of demoting coverage? Does a more comprehensive view of student learning get in the way of content knowledge? Recent research done by Marina Kogan and Sandra Laursen, brought to my attention by Yoshinobu and David Bressoud, suggests that students don’t necessarily suffer, and may be helped, from a holistic approach. From the conclusion to the Kogan and Laursen paper:
College instructors using student-centered methods in the classroom are often called upon to provide evidence in support of the educational benefits of their approach—an irony, given that traditional lecture approaches have seldom undergone similar evidence-based scrutiny. Our study indicates that the benefits of active learning experiences may be lasting and significant for some student groups, with no harm done to others. Importantly, “covering” less material in inquiry-based sections had no negative effect on students’ later performance in the major. Evidence for increased persistence is seen among the high-achieving students whom many faculty members would most like to recruit and retain in their department.
Still, it’s often difficult to prevent concerns about coverage from hijacking day-to-day teaching practice, regardless of course format. Here are some approaches I am using to keep coverage in perspective.
Regard conceptual understanding, mathematical writing and speaking, and other learning goals as integral parts of the “coverage” list, on an equal par with specific topics. Yoshinobu points out that we have a “systemic” issue, in that our institutions define coverage as no more than the list of topics. Hence I have to make a conscious effort, in planning each course, to weave all of the goals together, and to recognize that procedural skills won’t last without conceptual understanding, which in turn won’t happen if students don’t routinely speak and write mathematics.
Include learning objectives, not just a topics list, on the syllabus. Whether or not all of my students read the syllabus, it’s my way of formalizing my intentions and expectations. It’s also an invitation to consider the course in its entirety. This is especially important in mathematics, where students don’t understand many of the terms in a catalog description until after they’ve taken the course.
Have conversations with students, early and often, about the learning goals for the course. On the first day of linear algebra this semester, I devoted the entire hour to a class activity adapted from a model offered by Dana Ernst. The students’ responses to “What are the goals of a liberal arts education?” included “critical thinking” and “to experience the freedom to explore.” To “What can you reasonably expect to remember from your courses in 20 years?” I heard, “NOT details or the stuff you’re tested on,” but rather “how to figure out what’s relevant.” My own students understand the big picture; surely I can keep it in mind!
Halfway through the term, I had my students read this blog post from Ben Orlin and then fill out a survey online. I asked: to what extent are you practicing in the Church of Learning, as opposed to the Church of the Right Answer? Once again, the students reinforced my choices. Many of them also noted that their pre-college experiences, especially Advanced Placement Calculus, leaned heavily toward the Right Answer doctrine. In at least some cases, I’m working against students’ most recent experience of mathematics learning, so I need to be persistently transparent.
Gather data frequently on student understanding. Formative assessment isn’t just for elementary school teachers. I’m fortunate to teach small classes, so I can learn a lot just from classroom conversations. In an earlier post, I explained how recent research on learning has influenced my teaching. If I hear someone struggling to use “linearly independent” accurately during small group work, I can offer corrective feedback immediately. My students often show their work using a document projector. Anonymous surveys are useful as well; it only takes a few minutes for students to write down what’s puzzling them at the moment. I’ve never used clickers, but I’m intrigued by Eric Mazur’s methods. Most importantly, I try to design homework assignments that ask for deeper understanding. (It takes several weeks to convince students that homework is for formative, not summative, assessment, and that the graders’ job is to give constructive feedback.)
Bring student graders and teaching assistants in on the plan. I handpicked my graders this term, and made it clear that I want homework solutions to be clear and well-written, not just correct. They know that I’ve encouraged the students to show their attempts and partial solutions to more challenging problems. They let me know what misconceptions they see. The student tutors are also aware of my intentions.
It may be that I am especially sensitive to questions about coverage because my semester includes only twelve weeks of classes. My department colleagues and I agree that this poses a particularly vexing challenge in multivariable calculus. Getting to Green’s Theorem is challenging enough, and a thorough treatment of Stokes’ Theorem, which would add coherence to the entire semester, seems a worthy goal. Yet even here, I remind myself, what’s important is not only what I cover; it’s also what the students can retain.