Many of us who teach have had this experience: We work hard to explain to our students something that we understand well. We try to use intuition, analogies, examples, multiple methods, asking and answering questions, group exercises—the stuff of pedagogical knowledge. We are rewarded with students who feel that they understand. But when our students try to solve problems themselves, many make mistakes which reveal that they, in fact, didn’t understand. We correct their mistakes, explaining the right logic. But some students make the same mistakes again and again.
Through years of work and after much frustration, we teachers learn students’ common errors and the logic of those errors. We learn to stave the errors off—or use them as teaching moments. With our students, w go through each stage in their logic to find and explain the flaws.
This knowledge of the thinking behind their errors is not the content of our subject. Nor is it classic pedagogy. Most of us learn how our students misunderstand unsystematically and almost by accident. Some of us may systematically take stock of and analyze students’ mistakes to understand them. But in my experience few professors think of this as systematic knowledge and fewer yet have a name for it. I certainly did not.
Then a few weeks ago I read a New York Times magazine article, “Building a Better Teacher.” Researchers in elementary school math pedagogy, particularly Deborah Ball of the University of Michigan, have coined the term “Mathematics Knowledge for Teaching”. Here is how the article describes it:
“It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able to understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it… This was neither pure content knowledge nor what educators call pedagogical knowledge.”
The concept of mathematics knowledge for teaching is much broader than simply understanding misunderstandings. But I think that this sub-component is particularly valuable and something that we in higher education should embrace.
I’m sure some of you are already thinking that there is already a literature in subject-specific teaching in higher education. I have certainly seen and found valuable materials on the teaching of microeconomics and of statistics, two of the subjects that I teach. But in my experience, such materials combine content and pedagogy; they generally do not focus on understanding students’ misunderstandings.
The most efficient way to work on this is for instructors that teach the same subjects to collaborate, compare notes and so on. One semester, my colleague Gregg Van Ryzin and I both taught research methods with the same materials. Each week, we met to discuss what worked and what didn’t. We focused on the what, why and how of our students’ confusion. Our teaching and teaching materials improved substantially.
Unfortunately, I am still taking baby steps with such efforts. The real professionals know how to do this well. I learned recently about the work of Steve Hinds and others working on in developmental math in CUNY. Their work in general is described in “More Than Rules: College Transition Math Teaching for GED Graduates at the City University of New York”. What struck me most is how they work: all instructors collaborate on developing the materials but whatever the differences of opinion, everyone teaches with the same detailed materials, including in-class exercises, approaches for introducing topics and so on. Then all instructors describe their experiences: what worked, what didn’t, why and how. Collectively, they then work to improve student learning. This method, like the faculty inquiry groups Mary Taylor Huber has described, are very different from most of our experiences teaching in higher education.
What subject-specific student misunderstandings have you learned about? How has such knowledge helped your teaching? Do you and other professors who teach the same course regularly debrief?