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Author Archives: Dahlia Remler
Posts: 2 (archived below)
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?
When students are having trouble, I often suggest that they come to work with me one-on-one so that we can figure out where they are going wrong and fix it. We go through an example of a basic and core problem, such as a simple supply and demand problem in economics. I ask them to explain things to me step-by-step so that I can see at what point the difficulty or mistake occurs. When we get to a point where they don’t know how to answer a question or where they have a misunderstanding, I jump in and explain. Then we do another example—or several more—so that they have a chance to do it themselves. Students usually find where their misunderstanding, or missing understanding, is.
Some students feel upset at being “put on the spot,” and some just avoid coming in to do this. I know that it is important that students don’t think that they are innately stupid and can’t learn. And I worry that highlighting their lack of understanding can undermine students’ confidence. But this method of finding out what students don’t know is generally very effective. Afterwards, they frequently enjoy a major leap in performance and understanding.
One student response, however, undermines the process and raises a red flag: “I understood that” right after saying something that showed not understanding. If you don’t realize that you don’t understand something then you can’t fix it. In my experience, such a response indicates a student who is unlikely to learn and improve either from our interaction or in other settings. Over the years, I have come to repeat things like: “Knowing what you don’t know is the key to learning”; “In economics (research methods, etc.) many problems are hard, and I often don’t understand them at first. It takes work. You need to be comfortable with not understanding things. Not understanding doesn’t mean anything is wrong, just that you need to work at it.”
But these are things that I say primarily in my office with individual students and only rarely to the class as a whole. Even when I speak to the class as a whole, I don’t really offer concrete methods for helping a student who does not recognize what they don’t understand to gain that recognition. I would like to have a more systematic way to help in this process.
Therefore, I read with interest an account of a CUNY project on self-regulated learning, described recently in the Chronicle of Higher Education here . The article describes how in a basic math course at City Tech, “when students make errors, they need to be coached to reflect on exactly where they went wrong…students are required to rework at least two of their incorrect quiz problems…[and] write a sentence or two about the correct strategy.”
What do other instructors think? How do you respond to students who don’t recognize when they don’t know or understand something?