{"id":693,"date":"2015-04-20T00:01:39","date_gmt":"2015-04-20T04:01:39","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=693"},"modified":"2015-04-14T16:36:21","modified_gmt":"2015-04-14T20:36:21","slug":"taming-the-coverage-beast","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2015\/04\/20\/taming-the-coverage-beast\/","title":{"rendered":"Taming the Coverage Beast"},"content":{"rendered":"

By Priscilla Bremser, Contributing Editor<\/a>, Middlebury College<\/em><\/p>\n

By the end of every workshop and conference session on Inquiry-Based Learning that I\u2019ve attended, someone has raised a hand to ask about coverage. \u201cDon\u2019t you have to sacrifice coverage if you teach this way?\u201d Of course coverage took center stage for many of my professional conversations long before I tested the IBL waters; it\u2019s important. But an equally important question is this: What do we sacrifice when coverage dominates? It may well be conceptual understanding; it\u2019s possible to cover more ground, albeit thinly, if we settle for procedural understanding instead. More than once I\u2019ve settled for even less, delivering a quick lecture just so that my students will have \u201cseen\u201d 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?<\/p>\n

Strong arguments for striking that balance have been made elsewhere.\u00a0 For example, Stan Yoshinobu and Matthew Jones offer a close examination of the \u201cprice of coverage”<\/a>.\u00a0 “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. \u201cImplied messages are sent to students through classroom experiences,\u201d and some of those messages may have unproductive consequences, including overreliance on mimicking the instructor and memorization, and significant difficulties with non-routine problems.<\/p>\n

Is there, on the other hand, a price of demoting coverage?\u00a0 Does a more comprehensive view of student learning get in the way of content knowledge?\u00a0 Recent research done by Marina Kogan and Sandra Laursen<\/a>, brought to my attention by Yoshinobu<\/a> and David Bressoud<\/a>, suggests that students don\u2019t necessarily suffer, and may be helped, from a holistic approach. From the conclusion to the Kogan and Laursen paper:<\/p>\n

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\u2014an 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, \u201ccovering\u201d less material in inquiry-based sections had no negative effect on students\u2019 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.<\/p><\/blockquote>\n

Still, it\u2019s 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.<\/p>\n

Regard conceptual understanding, mathematical writing and speaking, and other learning goals as integral parts of the \u201ccoverage\u201d list, on an equal par with specific topics.<\/strong> Yoshinobu points out<\/a> that we have a \u201csystemic\u201d 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\u2019t last without conceptual understanding, which in turn won\u2019t happen if students don\u2019t routinely speak and write mathematics.<\/p>\n

Include learning objectives, not just a topics list, on the syllabus. <\/strong> Whether or not all of my students read the syllabus, it\u2019s my way of formalizing my intentions and expectations. It\u2019s also an invitation to consider the course in its entirety. This is especially important in mathematics, where students don\u2019t understand many of the terms in a catalog description until after they\u2019ve taken the course.<\/p>\n

Have conversations with students, early and often, about the learning goals for the course.<\/strong> On the first day of linear algebra this semester, I devoted the entire hour to a class activity adapted from a model<\/a> offered by Dana Ernst. The students\u2019 responses to \u201cWhat are the goals of a liberal arts education?\u201d included \u201ccritical thinking\u201d and \u201cto experience the freedom to explore.\u201d To \u201cWhat can you reasonably expect to remember from your courses in 20 years?\u201d I heard, \u201cNOT details or the stuff you\u2019re tested on,\u201d but rather \u201chow to figure out what\u2019s relevant.\u201d My own students understand the big picture; surely I can keep it in mind!<\/p>\n

Halfway through the term, I had my students read this blog post<\/a> 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\u2019m working against students\u2019 most recent experience of mathematics learning, so I need to be persistently transparent.<\/p>\n

Gather data frequently on student understanding<\/strong>. Formative assessment isn\u2019t just for elementary school teachers. I\u2019m fortunate to teach small classes, so I can learn a lot just from classroom conversations.\u00a0\u00a0 In an earlier post, I explained how recent research on learning has influenced my teaching<\/a>. If I hear someone struggling to use \u201clinearly independent” accurately during small group work, I can offer corrective feedback immediately. My students often show their work using a document projector.\u00a0 Anonymous surveys are useful as well; it only takes a few minutes for students to write down what’s puzzling them at the moment.\u00a0 I\u2019ve never used clickers, but I\u2019m intrigued by Eric Mazur<\/a>\u2019s 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\u2019 job is to give constructive feedback.)<\/p>\n

Bring student graders and teaching assistants in on the plan<\/strong>. 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\u2019ve 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.<\/p>\n

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\u2019s Theorem is challenging enough, and a thorough treatment of Stokes\u2019 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\u2019s also what the students can retain.<\/p>\n

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By Priscilla Bremser, Contributing Editor, Middlebury College By the end of every workshop and conference session on Inquiry-Based Learning that I\u2019ve attended, someone has raised a hand to ask about coverage. \u201cDon\u2019t you have to sacrifice coverage if you teach … Continue reading →<\/span><\/a><\/p>\n

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