"When good learning is happening"
Bob Megginson, a Thurnau Professor of Mathematics, believes that “a lot of the elements of good teaching come about from having the instructor be able to recognize when good learning is happening.” For good learning to happen, students have to be actively engaged, according to Megginson. While students often need their instructors to “give a good, well constructed description of how to do something,” they also need “to be invited to grapple” with those problems on a regular basis. This active grappling with problems allows students to “actually get it and internalize it, so they actually have a strong sense not just that it’s true, but why it’s true,” says Megginson.
Megginson explains that “one of the premier things that writing does for the student is that it forces them to clarify their own ideas.” Writing assignments help mathematics instructors gage what their students’ processes of reasoning are—not just whether they’ve arrived at the right answer. Megginson says that “the only way…to actually get that kind of information is to actually ask them to explain it to you. And if they’re going to explain it to you, first of all, I believe they have to explain it clearly, and concisely,and that requires organized thinking….It’s not just for that sort of assessment purpose, but it has actually pedagogical value.”
Concepts like the difference between a function and its derivative can often confuse students in math, Megginson explains. Graphs can be difficult to interpret, and the ways that mathematicians use language can throw students for a loop. So another way that Megginson gets students engaged in his calculus classroom is to put students into small problem-solving groups, “where they’re actually learning from each other and bouncing ideas off each other.” Collaborative learning makes it possible for students to engage not only with the material, but also with each other, he says. When students work together in the classroom, they also have an opportunity to stumble onto points of confusion and get clarification from the instructor.
Megginson believes that when teachers make themselves available to their students and regularly do formative assessment, they are more likely to notice when students aren’t comprehending something. In large lecture classrooms, they can use “clicker” technology to determine what percentage of students in the class have or have not understood any given concept. Because students learn in different ways, it is important for instructors to recognize the value of visual and experiential approaches, for instance. And instead of repeating the same lesson in the same way--which is unlikely to clarify material which students have had difficulty with-- “I can go at it from the other direction…until, with maybe another question or two that I ask, I’m confident that most of them have it,” Megginson says, “and then I can go on.”
Megginson uses concrete examples to illustrate concepts and relate learning to students’ own experiences. But he pays special attention to cultural assumptions that are often embedded in math problems. For instance, a question beginning, “An airplane flies over the football stadium and drops a sack of flour on the fifty yard line” will be more difficult for students who have never seen a football stadium. Megginson has found that collaborative groupwork can be a valuable way of exposing such assumptions and inviting students to learn from each other.
Finally, Megginson emphasizes how important both literacy and numeracy are for participation in our society today. He wants students to recognize how a decision made in the early grades to rule out the study of math or writing can narrow career choices. “When I see a student like this,” he says, “I wish that I could teleport myself back some number of years to when the student was taking those courses.” And Megginson points to legendary civil rights activist and mathematician, Robert Moses, who teaches in the UM School of Education, as a source of inspiration.